Two families of lines are given by $a x+b y+c=0$ and $4 a^2+9 b^2-c^2-12 a b=0$. Then, the line common to both the families is

Two non-parallel sides of a rhombus are parallel to the lines $x+y-1=0$ and $7 x-y-5=0$. If $(1,3)$ is the centre of the rhombus and one of its vertices $A(\alpha, \beta)$ lies on $15 x-5 y=6$, then one of the possible values of $(\alpha+\beta)$ is

If the equations $3 x^2+2 h x y-3 y^2=0$ and $3 x^2+2 h x y-3 y^2+2 x-4 y+c=0$ represent the four sides of a square, then $\frac{h}{c}=$

$(a, b)$ are the new coordinates of the point $(2,3)$ after shifting the origin to the point $(3,2)$ by translation of axes. If $(c, d)$ are the new coordinates of the point $(a, b)$ after rotating the axes through an angle $\frac{\pi}{4}$ about the origin in the anti-clockwise direction, then $d-c=$

The lines $x+y+4=0, x-2 y-4=0$ and $3 x+4 y-2=0$

The area of the triangle formed by the line $L$ with the coordinate axes is 12 sq. units. If $L$ passes through the point $(12,4)$ and the product $P$ of $X$ - intercept of $L$ and square of the $Y$-intercept of $L$ is negative, then $P=$

The area of the quadrilateral formed by the lines $x+2 y+3=0,2 x+4 y+9=0, x-2 y+3=0$ and $3 x-6 y+11=0$

If $(-1,-1)$ is the point of intersection of the pair of lines $2 x^2+5 x y-3 y^2+2 g x+2 f y+c=0$. Then $g+f$

A straight line passing through a point $(3,2)$ cuts $X$ and $Y$ axes at the points $A$ and $B$ respectively. If a point $P$ divides $A B$ in the ratio $2: 3$, then the equation of the locus of point $P$ is

By shifting the origin to the point $(-1,2)$ through translation of axes, if $a x^2+2 h x y+b y^2+2 g x+2 f y+c=0$ is the transformed equation of $2 x^2-x y+y^2-3 x+4 y-5=0$, then $2(f+g+h)=$

If a line $L$ passing through the point $A(-2,4)$ makes an angel of $60^{\circ}$ with the positive direction of $X$ - axis in anti-clockwise direction and $B(p, q)$ lying in the 3rd quadrant is a point on $L$ at the distance of 6 units from the point $A$, then $\sqrt{p^2+q^2-8 q}=$

If the perpendicular drawn from the point $(2,-3)$ to the straight line $4 x-3 y+8=0$ meets it at $M(a, b)$ and $a^3-b^3=k^3$, then $k=$

Let $Q$ be the image of a point $P(1,2)$ with respect to the line $x+y+1=0$ and $R$ be the image of $Q$ with respect to the line $x-y-1=0$. If $M$ and $N$ are the mid-points of $P Q$ and $Q R$ respectively, then $M N=$

If the slopes of the lines represented by the equation $6 x^2+2 h x y+4 y^2=0$ are in the ratio $2: 3$, then the value of $h$ such that both the lines make acute angles with the positive $X$-axis measured in positive direction is

If $2 x^2+x y-6 y^2+k=0$ is the transformed equation of $2 x^2+x y-6 y^2-13 x+9 y+15=0$ when the origin is shifted to the point $(a, b)$ by translation of axes, then $k=$

The line $L \equiv 6 x+3 y+k=0$ divides the line segment joining the points $(3,5)$ and $(4,6)$ in the ratio $-5: 4$. If the point of intersection of the lines $L=0$ and $x-y+1=0$ is $P(g, h)$, then $h=$

A straight line through the point $P(1,2)$ makes an angle $\theta$ with positive X -axis in anticlockwise direction and meets the line $x+\sqrt{3 y}-2 \sqrt{3}=0$ at $Q$. If $P Q=\frac{1}{2}$, then $\theta=$

The lines $x-2 y+1=0,2 x-3 y-1=0$ and $3 x-y+k=0$ are concurrent. The angle between the lines $3 x-y+k=0$ and $m x-3 y+6=0$ is $45^{\circ}$. If $m$ is an integer, then $m-k=$

If $\tan ^{-1}(2 \sqrt{10})$ is the angle between the lines $a x^2+4 x y-2 y^2=0$ and $a \in Z$, then the product of the slopes of given lines is

The point $P(\alpha, \beta)(\alpha>0, \beta>0)$ undergoes the following transformations successively.

(a) Translation to a distance of 3 units in positive direction of $X$-axis.

(b) Reflection about the line $y=-x$.

(c) Rotation of axes through an angle of $\frac{\pi}{4}$ about the origin in the positive direction.

If the final position of that point $P$ is $(-4 \sqrt{2},-2 \sqrt{2})$, then $(\alpha+\beta)=$

If the line passing through the point $(4,-3)$ and having negative slope makes an angle of $45^{\circ}$ with the line joining the points $(1,1),(2,3)$, then the sum of intercepts of that line is

$O(0,0), B(-3,-1)$ and $C(-1,-3)$ are vertices of a $\triangle O B C$. $D$ is a point on $O C$ and $E$ is a point on $O B$. If the equation of $D E$ is $2 x+2 y+\sqrt{2}=0$, then the ratio in which the line $D E$ divides the altitude of the $\triangle O B C$ is

Every point on the curve $3 x+2 y-3 x y=0$ is the centroid of a triangle formed by the coordinate axes and a line $(L)$ intersecting both the coordinates axes. Then, all such lines $(L)$

The value of ' $a$ ' for which the equation $\left(a^2-3\right) x^2+16 x y -2 a y^2+4 x-8 y-2=0$ represents a pair of perpendicular lines is

If the points $A(2,3), B(3,2)$ form a triangle with a variable point $p\left(t, t^2\right)$, where $t$ is a parameter, then the equation of the locus of the centroid of $\triangle A B C$ is

If $(h, k)$ is the new origin to be chosen to eliminate first degree terms from the equation $S \equiv 2 x^2-x y-y^2-3 x+3 y=0$ by translation and if $\theta$ is the angle with which the axes are to be rotated about the origin in anti-clockwise direction to eliminate $x y$-term from $S=0$, then $\tan 2 \theta=$

A line $L$ perpendicular to the line $5 x-12 y+6=0$ makes positive intercept on the $Y$-axis. If the distance from the origin to the line $L$ is 2 units and the angle made by the perpendicular drawn from the origin to the line $L$ with positive $X$-axis is $\theta$, then $\tan \theta+\cot \theta=$

If a line $L$ passing through a point $A(2,3)$ intersects another line $4 x-3 y-19=0$ at the point $B$ such that $A B=4$, then the angle made by the line $L$ with positive $X$-axis in anti-clockwise direction is

A variable straight-line $L$ with negative slope passes through the point $(4,9)$ and cuts the positive coordinate axes in $A$ and $B$. If $O$ is the origin, then the minimum value of $O A+O B$ is

If $4 x^2+12 x y+9 y^2+2 g x+2 f y-1=0$ represent a pair of parallel lines, then

A straight line passing through a fixed point $(-3,4)$ intersects the coordinate axes at $A$ and $B$. If $O$ is the origin and $O A B C$ forms a rectangle, then the locus of $C$ is

When the origin is shifted to the point $P$ by translation of axes, the equation $2 x^2+y^2-4 x+4 y=0$ is transformed to $2 x^2+y^2-8 x+8 y+18=0$. Then, the transformed equation of the straight line $x+2 y+2=0$, if the origin is shifted to the same point $P$ is

If the lines $x+y-1=0, k x+2 y+1=0$ and $4 x+2 k y+7=0$ are concurrent, then $k=$

If $\alpha, \beta(\alpha>\beta)$ are two values of $k$ such that the equations $2 x+(3-2 k) y+(2 k+1)=0$ and $k x+(k-1) y-4=0$ represents two perpendicular lines, then $\alpha^2+2 \beta=$

If $k=\frac{a+b}{a b}$ is a non-zero constant, then the point which lies on the straight line $\frac{x}{a}+\frac{y}{b}=1$ is

The point of concurrence of all the chords of the curve $3 x^2-y^2-2 x+4 y=0$ which subtend a right angle at the origin is

Let $d$ be the distance between the parallel lines $3 x-2 y+5=0$ and $3 x-2 y+5+2 \sqrt{13}=0$.

Let $L_1 \equiv 3 x-2 y+k_1=0\left(k_1>0\right)$ and $L_2 \equiv 3 x-2 y+k_2=0\left(k_2>0\right)$ be two lines that are at the distance of $\frac{4 d}{\sqrt{13}}$ and $\frac{3 d}{\sqrt{13}}$ from the line $3 x-2 y+5=0$.

Then, the combined equation of the lines $L_1=0$ and $L_2=0$ is

If $(h, k)$ is the image of the point $(3,-4)$ with respect to the line $2 x-3 y-5=0$ and $(l, m)$ is the foot of the perpendicular from $(h, k)$ on to the line $3 x+2 y+12=0$, then $l h+m k+1=$

A straight line parallel to the line $y=\sqrt{3} x$ passes through $Q(2,3)$ and cuts the line $2 x+4 y-27=0$ at $P$. Then, the length of the line segment $P Q$ is

If a line $a x+2 y=k$ forms a triangle of area 3 sq. units with the coordinate axis and is perpendicular to the line $2 x-3 y+7=0$, then the product of all the possible values of $k$ is

The orthocenter of the triangle whose sides are given by $x+y+10=0, x-y-2=0$ and $2 x+y-7=0$ is

For $l \in R$, the equation $(2 l-3) x^2+2 l x y-y^2=0$ represents a pair of distinct lines

Two points $P(a, 2)$ and $Q(1, b)$ lie on either side of the line $2 x-3 y+1=0$. If $P$ is the point of intersection of the lines $4 x+3 y+k=0$ and $3 x+4 y+k=0$, then the range of $b$ is

Let the angle between the lines $x-2 y+3=0$ and $k x-y+2=0$ be $45^{\circ}$. If $k_1, k_2\left(k_1>k_2\right)$ are two distinct real values of $k$, then $k_1-2=$

If the lines $4 x+3 y-k=0,2 x+y+3=0$ and $3 x+2 y+k=0$ are concurrent, then the perpendicular distance from the point of concurrency of these lines to the line $3 x+4 y+2=0$ is

Let $A(1,3)$ and $B(2,5)$ be two points and $C(h, k)$ be a point such that $B C$ is perpendicular to $A C$. If $\angle C A B=\angle C B A$, then $h=$

Let the line $2 x-3 y-1=0$ intersect the curve $x^2+2 x y+5 y^2+2 x+3 y-1=0$ in distinct points $A$ and $B$. If ' $O$ ' is the origin, then $\cos \angle A O B=$

If $\alpha$ is the angle made by the perpendicular drawn from origin to the line $3 x-4 y+5=0$ with positive $X$-axis in positive direction and $a x+b y=1$ is the equation of a line passing through the point $(1,-1)$ with $\tan \alpha$ as its slope, then $a+a b+b=$

If $L_1$ is a line passing through the point $P(4,-3)$ and perpendicular to the line $3 x-4 y+k=0$ then the distance of $P$ from the line $5 x-3 y-2=0$ measured along the line $L_1$ is

Let the line $L_1$ passing through the point of intersection of the lines $2 x+3 y-5=0$ and $4 x-5 y+7=0$ divide the line segment joining the points $(2,3)$ and $(1,-1)$ in the ratio $2: 1$. If the equation of $L_1$ is $a x+b y=1$, then $33(a-b)=$

Let $A B C$ be a triangle and $A=(1,2)$. If $x-3 y-5=0$ the and $x+5 y-9=0$ are the perpendicular bisectors of the sides $A B$ and $B C$ respectively, then the length of the side $A C$ is

Let $A(4,3,5), B(1,-2,1), C(3,2,1)$ be the vertices of a $\triangle A B C$. If the internal bisector of $\angle B A C$ meet the side $B C$ at $D$, then $C D=$

A line $L$ has intercepts $a$ and $b$ on the coordinate axes. When the coordinate axes are rotated through an angle $\alpha$ and keeping the origin fixed, the same line $L$ has intercepts $p$ and $q$ on the new axes. Then,

Two lines $L_1$ and $L_2$ passing through the point $P(1,2)$ cut the line $x+y=4$ at a distance of $\frac{\sqrt{6}}{3}$ units from $P$. Then, the angles made by $L_1, L_2$ with positive $X$-axis are

A pair of straight lines drawn though the origin forms. an isosceles triangle right angled at the origin with the line $2 x+3 y=6$. The area (in sq units) of the triangle, so formed is

The equation of the straight line passing through the point $(3,2)$ and inclined at an angle of $60^{\circ}$ with the line $\sqrt{3} x+y=1$ is

An equilateral triangle is constructed between the lines $\sqrt{3} x+y-6=0$ and $\sqrt{3} x+y+9=0$ with base on one line and vertex on the other. The area (in sq units) of the triangle, so formed is

If $\theta$ is the acute angle between the lines joining the origin to the points of intersection of the curve $x^2+x y+y^2+x+3 y+1=0$ and the straight line $x+y+2=0$, then $\cos \theta=$

$A(-4,0)$ and $B(4,0)$ are two fixed points. $C$ and $D$ are two points on $Y$ - axis such that $C D=4$ and $C$ is a point below $D$. Then, the locus of the point of intersection of the lines $A C$ and $B D$ is

By rotating the axes through an angle of $30^{\circ}$ in the anti-clockwise direction about the origin, the equation $4 x^2+12 x y+9 y^2+6 x+9 y+2=0$ becomes $a x^2+2 h x y+b y^2+2 g x+2 f y+c=0$ becomes, then

In an isosceles triangle the ends of its base are $(2 a, 0),(0, a)$ and one of its two other sides is a horizontal line other than $X$-axis. If the third vertex is $\left(x_1, y_1\right)$, then $x_1+y_1=$

If the lines $L_1 \equiv x-2 y+3=0, L_2 \equiv 2 x+y+1=0$ and $L_3 \equiv 3 x+y+c=0$ are concurrent and $\theta$ is the acute angle between the lines $L_1=0$ and $L_3=0$, then $\tan \theta=$

If the lengths of the perpendiculars drawn from a point $(a, b)$ to the lines $2 x+3 y+4=0$ and $3 x-2 y+4=0$ are same, then the point $(a, b)$ lies on the line

If $3 x+6 y+2=0, x+y+1=0,2 x-y+3=0$ are three given lines, then the point $\left(\frac{-4}{3}, \frac{1}{3}\right)$ is

If $\theta$ is the acute angle between the pair of lines $12 x^2+2 h x y+7 y^2=0$ and $\tan \theta=\frac{8}{19}$, then $h=$

The number of real values of $\alpha$ for which the pair of lines represented by $\left(\alpha^2+12|\alpha|\right) x^2+6 x y+(18-21|\alpha|) y^2=0$ are at right angles to each other, is

The line $x+2 y-c=0$ meets the curve $x^2+y^2-3 x-6 y+3=0$ at two points $P$ and $Q$ and $\angle P O Q=\frac{\pi}{2}$, where $O$ is the origin. Then, $2 c^2-15 c=$

Let $A B C$ be a triangle. Let a point $P$ divide $A B$ in the ratio $1: 2$ internally and a point $Q$ divide $B C$ in the ratio $1: 2$ internally. Let $D$ be the point of intersection of $A Q$ and $C P$. If the area of the $\triangle A B C$ is $k$ square units, then the area of the $\triangle B C D$ in (sq. units) is

$B(2,3), C(5,-2), D(1,-1)$ are three points. If $A$ is a variable point such that the area of the quadrilateral $A B C D$ is 10 sq. units, then the locus of $A$ is

A line makes intercepts 5 and 7 on the coordinate axes. The axes are rotated through an angle $\theta$ in the positive direction about the origin so that the line makes equal intercepts on the new axes, then $|\tan \theta|=$

$L \equiv 7 x-y+8=0$ is one of the diagonals of a square for which $(-4,5)$ and $(3,4)$ are two vertices. Then, the coordinates of the two vertices lying on the diagonal $L=0$ are

The locus of the image of a variable point $(\alpha, 2 \alpha-1)$ with respect to the line $3 x-2 y+4=0$, is

Let $M$ be the foot of the perpendicular drawn from the point $(5,-7)$ to the line $3 x-5 y+1=0$. Then, the perpendicular distance from $M$ to the line $2 x+5 y-3=0$ is

If $P$ is a point equidistant from all the vertices $A(-1,3), B(3,5), C(5,7)$ of a $\triangle A B C$, then $P A=$

4 different pairs of lines are given in List I and the cosine of the angle between every pair of lines is given in List II. Match the following :

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If $a x^2+6 x y-2 y^2=0$ represents a pair of perpendicular lines and $9 x^2+2 h x y+4 y^2=0(h>0)$ represents a pair of coincident lines, then $h=$

The line $x+2 y=k$ meets the curve $2 x^2-2 x y+3 y^2+2 x-y-1=0$ at two points $A$ and $B$. Let $O$ be the origin. If the line segments $O A$ and $O B$ are perpendicular to each other, then $k=$

If a straight line $L$ passing through the point $(5,-3)$ is inclined at an angle of $60^{\circ}$ to the line $\sqrt{3} x+y-9=0$ and $L$ intersects $X$-axis, then the equation of $L$ is

Let $\alpha, \beta$ and $\gamma$ be three non-zero real constants and $a, b$ and $c$ be three arbitrary real numbers which satisfy $\alpha a+\beta b+\gamma c=0$. Then, the point of concurrence of the family of lines $a x+b y+c=0$ is

If the algebraic sum of the perpendicular distances from the points $(2,0),(0,2)$ and $(1,1)$ to a variable line is zero, then the variable line always passes through a fixed point. The coordinates of that point are

For $a, b, c \in R$, if $6 a^2-3 b^2-c^2+7 a b-a c+4 b c=0$ and $|a|+|b| \neq 0$, then all the lines given by $a x+b y+c=0$ are

If $\theta$ is the acute angle between the pair of lines $H \equiv a x^2-x y+b y^2=0, \tan \theta=5$ and $(1,-1)$ is a point on $H=0$, then $a^2+a b+b^2=$

The equation of the pair of straight lines passing through the point $(2,3)$ and perpendicular to the pair of lines $3 x^2-4 x y+5 y^2=0$ is $a x^2+2 h x y+b y^2+2 g x+2 f y+c=0$, then $a+b+c+f+g+h=$

If $f(x, y)=0$ is the combined equation of the lines joining the origin to the points where the line $4 x-6 y-2=0$ meets the curve $3 x^2-4 x y+5 y^2-2 x+y-6=0$, then $\frac{f(1,-1)}{f(-1,-1)}=$

If the line $2 x-y-4=0$ divides the line segment joining the points $(2,-1)$ and $(1,-4)$ at the point $(a, b)$ in the ratio $m: n$, then $4\left(a-b\left(\frac{m}{n}\right)^2\right)=$

The distance between the points of concurrency of the two families of straight lines given by $x+(5 \lambda+1) y+1-3 \lambda=0$ and $(5 \mu+2) x-3 y+3+6 \mu=0$ is

Let the line $L$ drawn perpendicular to the lines $2 x-3 y+4=0$ and $6 x-9 y+7=0$ meet them at $A$ and $B$, respectively. If $P(\mathrm{l}, \mathrm{l})$ is a point on $L$, then the ratio in which $P$ divides $A B$ is

The orthocentre of the triangle formed by the points $(1,3),(-3,5)$ and $(5,-1)$ is

If $\alpha x^2+2 \gamma x y+\beta y^2=0$ is the equation of pair of lines passing through the origin and perpendicular to the pair of lines $b h x^2+a b x y+a h y^2=0(a \neq 0, b \neq 0)$, then $\alpha \beta / \gamma^2=$

$\frac{x^2}{a}+\frac{x y}{h}+\frac{y^2}{b}=0(a \neq 0, h \neq 0, b \neq 0)$ represents two coincident lines if

If the lines joining the origin to the points of intersection of the line $x+y=k$ and the curve $x^2+y^2-2 x-4 y+2=0$ are at right angles, then the sum of all the possible values of $k$ is

The transformed equation of $3 X^2+4 X Y+Y^2-8 X-4 Y-4=0$ is $f(X, Y)=a X^2+2 h X Y+b Y^2+c=0$ when the origin is shifted to a new point by the translation of axes. Then, $f(1,1)=$

If the line $2 x-3 y+4=0$ divides the line segment joining the points $A(-2,3)$ and $B(3,-2)$ in the ratio $m: n$, then the point which divides $A B$ in the ratio $-4 m: 3 n$ is

If the lines $L_1 \equiv 2 x+y+3=0, L_2 \equiv k x+2 y-3=0$ and $L_3 \equiv 3 x-2 y+1=0$ are concurrent then the cosine of the acute angle between the lines $L_2=0$ and $2 x-5 y+7=0$ is

If $Q$ is the image of the point $P(1,1)$ with respect to the straight line $x+y+1=0$, then the length of the perpendicular drawn from $Q$ to the line $3 x-4 y+3=0$ is

The centroid of the triangle formed by the lines $x-3 y+3=0, x+3 y+3=0 x+y-1=0$ is

If the slope of one of the lines represented by $5 x^2+\frac{40}{3} x y+k y^2=0$ is 3 , then the angle between the pair of lines is

If a line $L$ is common to the pairs of lines $6 x^2-x y-12 y^2=0$ and $15 x^2+14 x y-8 y^2=0$ then the combined equation the other two lines is

If $L$ is a line passing through the point $(-1,1)$ and parallel to the common line of the pairs of lines $6 x^2-x y-12 y^2=0$ and $15 x^2+14 x y-8 y^2=0$, then the equation of pair of lines joining the origin to the points of intersection of the curve $2 x^2-x y-y^2+x-y=0$ and the line $L$ is

Let $A(5,-3), B(3,-2), C(-1,5)$ be three points. If $P$ is a point satisfying the condition $P A^2+2 P B^2=3 P C^2$, then a point that lies on the locus of $P$ is

If $\theta$ is the acute angle between the lines $\frac{x}{a}+\frac{y}{b}=1, \frac{x}{b}+\frac{y}{a}=1$, then $\sin \theta=$

If the line $x-y+1=0$ cuts the lines $2 x+2 y+3=0$ and $3 x+3 y+2=0$ at the points $A$ and $B$ respectively, then $A B=$

If the incentre and the circumcentre of the triangle formed by the lines $x=2,4 x+3 y+7=0$ and $y=3$ are $I$ and $S$ respectively, then $I S=$

$a x^2-4 x y-2 y^2=0$ represents a pair of lines. If $\theta$ is the angle between these lines, $\cos \theta=\frac{1}{5}$ and the possible values of ' $a$ ' are $a_1$ and $a_2\left(a_1

Let $L_1, L_2$ be the lines represented by the equation $4 x^2-5 x y+3 y^2=0$. Let $L_3, L_4$ be two lines passing through the point $(4,3)$ such that $L_3$ and $L_4$ are perpendicular to $L_1$ and $L_2$ respectively. If the combined equation of $L_3$ and $L_4$ is $a x^2+2 h x y+b y^2+2 g x+2 f y+c=0$, and $a f+b g+c h=$

The equation $x^2-y^2+a x+b=0$ represents a pair of lines for the ordered pair $(a, b)=$

When the coordinate axes are rotated through an angle $\theta$ in anti clockwise direction, if the transformed equation of $x^2+y^2+2 x y+2 x+6 y+1=0$ is $(2+\sqrt{3}) X^2+2 X Y+(2-\sqrt{3}) Y^2+a X+b Y+2=0$, then $3 a-b=$

If the lines $3 x+y-4=0, x-a y-10=0, b x+2 y+9=0$ form three successive sides of a rectangle in that order and the fourth side passes through $(1,2)$, then the area of that rectangle (in sq. units) is

The points $A(2,1), B(3,-2)$ and $C(a, b)$ are vertices of the rectangle $A B C D$. If the point $P(3,4)$ lies on $C D$ produced, then $5 a+10 b=$

If $\left|\begin{array}{lll}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{array}\right|=0$, then the lines $a_i x+b_i y+c_i=0$

( $i=1,2,3$ ) represent

For integer $k$, if the area of the triangle formed by the pair of lines $S=3 x^2-2 k x y+y^2=0$ with the line $L=2 x-y-6=0$ is 36 sq. units, then for the angle $\theta$ between the lines $S=0, \sin \theta=$

If the sides of a triangle $A B C$ are $2 x^2-y^2=0$, $x+y-1=0$ and the sides of another triangle $P Q R$ are $2 x^2-5 x y+2 y^2=0,7 x-2 y-12=0$, then the distance between the centroid of $\triangle A B C$ and the orthocentre of $\triangle P Q R$ is

Let $A=(2,3), B=(3,-5)$ be two vertices of $\triangle A B C$ such that $C$ is a point on the line $L \equiv 3 x+4 y-5=0$. Then the locus of the centroid of $\triangle A B C$ is a line parallel to

If the normal form of the equation of a straight line $4 x+3 y+2=0$ is $x \cos \alpha+y \sin \alpha=p$ and its intercept form is $\frac{x}{a}+\frac{y}{b}=1$, then $\frac{p \sec \alpha}{a b}=$

For an integer $K$, if the point $P\left(K^2, K+1\right)$ and the origin $O(0,0)$ lie in the same region between the lines $x+2 y-5=0$ and $3 x-y+1=0$, then the possible number of such points $P$ is

The area (in square units) of the quadrilateral formed by the point of intersection of the lines $x+y-1=0$, $x-y+1=0$, the point $(1,1)$ and the feet of the perpendiculars from this point on to the lines is

The condition that the lines joining the origin to the points of intersection of the two curves $x^2+y^2+g x+c=0, x^2+y^2+2 f y-c=0$ are at right angles, is

If $\alpha$ represent the square of the distance between the origin and the point of intersection of the lines $x^2-y^2-x+3 y-2=0$ and $\beta$ represent the product of the perpendicular distances from the origin on the pair of lines, then $\alpha \beta=$

Let $A(2,1)$ be a point and equation of the straight line $L$ be $x-y=0$. Let $a$ and $b$ respectively represent the distances from a variable point $P(\alpha, \beta)$ to $A$ and to the line $L$. If $C$ is distance of the point $A$ from origin such that $a=b c$, then locus of $P$ is

The point $(4,1)$ undergoes the following transformations successively :

(i) Reflection is the line $x-y=0$

(ii) Shifting through a distance of 2 units along the positive $X$-axis

(iii) Projection on $X$-axis

The coordinates of the point in its final position are

Two straight lines $3 x+4 y=5$ and $4 x-3 y=15$ intersect at the point $A$. The equations of the lines passing through $(1,2)$ and intersecting the given lines at $B$ and $C$ such that $A B=A C$ are

The equation of a line making an angle $60^{\circ}$ with the line $x+y-3=0$ and passing through the point $(1,1)$ is

Let $P$ be the pair of lines represented by $2 x^2-5 x y+2 y^2+6 x-3 y=0$ and consider the following independent statements

(i) $\alpha$ is the $x$ coordinate of the point of intersection of the pair of lines $P$.

(ii) $\beta$ is the slope of one of the lines of $P$ passing through origin.

(iii) $\gamma$ is the constant term in the equation of the pair of angular bisectors of $P$.

Then,

The combined equation of the diagonals of the parallelogram formed by the lines

$ \left(7 x^2-4 x y+8 y^2\right)^2+(4 x-8 y-32)\left(7 x^2-4 x y+8 y^2\right)=0 $

is

If $M$ is the foot of the perpendicular drawn from the origin $O$ on to the variable line $L$, passing through a fixed point $(a, b)$, then the locus of the mid-point of $O M$ is

When the origin is shifted to the point $\left(\frac{3}{2}, \frac{3}{2}\right)$ by the translation of coordinate axes, then the transformed equation of $32 x^2+8 x y+32 y^2-108 x-108 y+99=0$ is

A line $L_1$ passing through $A(3,4)$ and having slope 1 cuts another line $L_2$ passing through $C$ at $B$, such that $A B=A C$. If the equation of line $B C$ is $2 x-y+4=0$, then the equation of $A C$ is

Angles made with the $X$-axis by the two lines passing through the point $P(1,2)$ and cutting the line $x+y=4$ at a distance $\frac{\sqrt{6}}{3}$ units from the point $P$ are

The straight lines $x+3 y-9=0,4 x+5 y-1=0$, $p x+q y+10=0$ are concurrent, if the line $5 x+6 y+10=0$ passes through the point

The straight lines $x+3 y-9=0,4 x+5 y-1=0$, $p x+q y+10=0$ are concurrent, if the line $5 x+6 y+10=0$ passes through the point

The centroid of the triangle formed by the lines $x+y=1$ and $2 y^2-x y-6 x^2=0$ is