The points $(2,3)$ and $\left(-4,-\frac{4}{3}\right)$ lie on the opposite sides of the line $L \equiv 5 x-6 y+k=0$ and k is an integer. If the points $(1,2)$ and $(4,5)$ lie on the same side of the line $L=0$, then the perpendicular distance from origin to the line $L=0$ is

If the incentre of the triangle formed by the lines $x-2=0, x+y-1=0, x-y+3=0$ is $(\alpha, \beta)$, then $\beta=$

If the equation of the pair of straight lines intersecting at ( $a, b$ ) and perpendicular to the pair of lines $3 x^2-4 x y+5 y^2=0$ is $l x^2+2 n x y+m y^2-32 x-26 y+c=0$, then $\frac{a+b+c}{l+h+m}=$

$P Q R$ is a right-angled isosceles triangle with right angle at $P(2,1)$. If the equation of the line $Q R$ is $2 x+y=3$, then the equation representing the pair of lines $P Q$ and $P R$ is

The coordinate axes are rotated about the origin in the counter clockwise direction through an angle $60^{\circ}$. If a and $b$ are the intercepts made on the new axes by a straight line whose equation referred to the original axes is $x+y=1$, then $\frac{1}{a^2}+\frac{1}{b^2}=$

The image of a point $(2,-1)$ with respect to the line $x-y+1=0$ is

If a straight line is at a distance of 10 units from the origin and the perpendicular drawn from the origin to it makes an angle $\frac{\pi}{4}$ with the negative $X$-axis in the negative direction, then the equation of that line is

If one of the lines given by the pair of lines $3 x^2-2 y^2+a x y=0$ is making an angle $60^{\circ}$ with $X$-axis, then $a=$

$A$ straight line passing through the origin $O$ meets the parallel lines $4 x+2 y=9$ and $2 x+y+6=0$ at the points $P$ and $Q$ respectively. Then, the point $O$ divides the line segment $P Q$ in the ratio

If the axes are translated to the orthocentre of the triangle formed by the points $\mathrm{A}(7,5), \mathrm{B}(-5,-7)$ and $C(7,-7)$, then the coordinates of the incentre of the triangle in the new system are

The angle made by a line $L$ with positive $X$-axis measured in the positive direction is $\frac{\pi}{6}$ and the intercept made by $L$ on $Y$-axis is negative. IF $L$ is at a distance of 5 units from the origin, then the perpendicular distance from the point $(1,-\sqrt{3})$ to the line $L$ is

$L_1$ and $L_2$ are two lines having slopes 2 and $-\frac{1}{2}$ respectively. If both $L_1$ and $L_2$ are concurrent with the lines $x-y+2=0$ and $2 x+y+3=0$, then sum of the absolute values of the intercepts made by the lines $L_1$ and $L_2$ on the coordinate axes is

The lines $L_1: y-x=0$ and $L_2: 2 x+y=0$ intersect the line $L_3: y+2=0$ at $P$ and $Q$ respectively. The bisector of the angle between $L_1$ and $L_2$ divides the line segment $P Q$ internally at $R$.

Statement $I P R: R Q=2 \sqrt{2}: \sqrt{5}$

Statement II In any triangle, bisector of an angle divides that triangle into two similar triangles

If $2 x^2+3 x y-2 y^2-5 x+2 f y-3=0$ represents a pair of straight lines, then one of the possible values of $f$ is

The point $P(4,1)$ undergoes the following transformations in succession :

(i) origin is shifted to the point $(1,6)$ by translation of axes.

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

(iii) rotation of axes through an angle of $90^{\circ}$ in the positive direction.

Then, the coordinates of the point $P$ in its final position are

$L_1 \equiv a x-3 y+5=0$ and $L_2 \equiv 4 x-6 y+8=0$ are two parallel lines. If $p, q$ are the intercepts made by $L_1=0$ and $m, n$ are the intercepts made by $L_2=0$ on the $X$, $Y$-coordinate axes respectively, then the equation of the line passing through the points $(p, q)$ and $(m, n)$ is

If $(h, k)$ is the image of the point $(2,-3)$ with respect to the line $5 x-3 y=2$, then $h+k=$

If the pair of lines $a x^2-7 x y-3 y^2=0$ and $2 x^2+x y-6 y^2=0$ have exactly one line in common and ' $a$ ' is an integer, then the equation of the pair of bisectors of the angles between the lines $a x^2-7 x y-3 y^2=0$ is

If the angle between the pair of lines $2 x^2+2 h x y+2 y^2-x+y-1=0$ is $\tan ^{-1}\left(\frac{3}{4}\right)$ and $h$ is a positive rational number, then the point of intersection of these two lines is

If $\alpha$ is the angle made by the perpendicular drawn from origin to the line $12 x-5 y+13=0$ with the positive $X$-axis in anti-clockwise direction, then $\alpha=$

If the equation of the pair of lines passing through $(1,1)$ and perpendicular to the pair of line $2 x^2+x y-y^2-x+2 y-1=0$ is $a x^2+2 h x y+b y^2+2 g x+3 y=0$, then $\frac{b}{a}=$

If the combined equation of the lines joining the origin to the point of intersection of the curve $x^2+y^2-2 x-4 y+2=0$ and the line $x+y-2=0$ is $\left(l_1 x+m_1 y\right)\left(l_2 x+m_2 y\right)=0$, then $l_1+l_2+m_1+m_2=$

Let $A(5,4)$ and $B(5,-4)$ be two points.

If $P$ is a point in the coordinate plane such that $\sqrt{A P B}=\frac{\pi}{4}$, then the point $P$ lies on the curve

If the perpendicular distances from the points $(2,3)$, $(4, a)$ and $(\alpha, \beta)$ on to the line $3 x+4 y-3=0$ are equal and $4 \alpha-3 \beta+1=0$, then sum of all possible values of $a, \alpha$ and $\beta$ is

The equation of the base of an equilateral triangle is $x+y=2$ and its opposite vertex is $(2,1)$. If $m_1, m_2$ are the slopes of the other two sides and the length of its side is $a$, then $\left|m_1-m_2\right|+a \sqrt{2}=$

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

If $A(1,0), B(0,-2)$ and $C(2,-1)$ are three fixed points, then the equation of the locus of a point $P$ such that area of $\triangle P A B$ is equal to area of $\triangle P A C$ is

The transformed equation of $3 x^2-4 x y=r^2$ when the coordinate axes are rotated about the origin through an angle of $\tan ^{-1}(2)$ in positive direction is

A line $L_1$ passing through the point of intersection of the lines $x-2 y+3=0$ and $2 x-y=0$ is parallel to the line $L_2$. If $L_2$ passes through origin and also through the point of intersection of the lines $3 x-y+2=0$ and $x-3 y-2=0$, then the distance between the lines $L_1$ and $L_2$ is

If the lines $x+y-2=0,3 x-4 y+1=0$ and $5 x+k y-7=0$ are concurrent at $(\alpha, \beta)$, then equation of the line concurrent with the given lines and perpendicular to $k x+y-k=0$ is

If two sides of a triangle are represented by $3 x^2-5 x y+2 y^2=0$ and its orthocentre is $(2,1)$, then the equation of the third side is

If $a x^2+2 h x y-2 a y^2+3 x+15 y-9=0$ represents a pair of lines intersecting at $(1,1)$, then $a h=$

A straight line passing through a fixed point $(2,3)$ intersects the coordinate axes at points $P$ and $Q$. If $O$ is the origin and $R$ is a variable point such that $O P R Q$ is a rectangle, then the locus of $R$ is

If the lines $x+2 a y+a=0, x+3 b y+b=0$, $x+4 c y+c=0$ are concurrent, then $a, b, c$ are in

If $M$ is the foot of the perpendicular drawn from the origin to the line $x-2 y+3=0$ which meets the $X$ and $Y$-axes at $A$ and $B$, respectively, then $A M=$

One line of the pair of lines $x^2+x y-2 y^2=0$ is perpendicular to one line of the pair of lines $3 y^2-5 x y-2 x^2=0$ If the combined equation of the two lines other than those two perpendicular lines is $a x^2+2 h x y+b y^2=0$, then $a+2 h+b=$

If the angle between the lines joining the origin to the points of intersection of $x+2 y+\lambda=0$ and $2 x^2-2 x y+3 y^2+2 x-y-1=0$ is $\frac{\pi}{2}$, then a value of $\lambda$. is

If $P$ is a variable point which is at a distance of 2 units. from the line $2 x-3 y+1=0$ and $\sqrt{13}$ units from the point $(5,6)$, then the equation of the locus of $P$ is

If the equation $3 x^2+4 y^2-x y+k=0$ is the transformed equation of $3 x^2+4 y^2-x y-5 x-7 y+2=0$ after shifting the origin to the point $(\alpha, \beta)$ by the translation of axes, then $\alpha+\beta-k=$

If the intercept of a straight line $L$ made between the straight lines $5 x-y-4=0$ and $3 x+4 y-4=0$ is bisected at the point $(1,5)$, then the equation of $L$ is

$A$ line $L$ passes through the point $P(1,2)$ and makes an angle of $60^{\circ}$ with $O X$ in the positive direction. $A$ and $B$ are two points lying on $L$ at a distance of 4 units from $P$. If $O$ is the origin, then the area of $\triangle O A B$ is

The equation $(2 p-3) x^2+2 p x y-y^2=0$ represents a pair of distinct lines

If the distance of a variable point $P$ from a point $A(2,-2)$ is twice the distance of $P$ from $Y$-axis, then the equation of locus of $P$ is

If the transformed equation of the equation $2 x^2+3 x y-2 y^2-17 x+6 y+8=0$ after translating the coordinate axes to a new origin ( $\alpha, \beta$ ) is $a X^2+2 h X Y+b Y^2+c=0$, then $3 \alpha+c=$

$P(6,4)$ is a point on the line $x-y-2=0$. If $A(\alpha, \beta)$ and $B(\gamma, \delta)$ are two points on this line lying on either side of $P$ at a distance of 4 units from $P$, then $\alpha^2+\beta^2+\gamma^2+\delta^2=$

If the straight line $2 x+3 y+1=0$ bisects the angle between two other straight lines one of which is $3 x+2 y+4=0$, then the equation of the other straight line is

If the slope of both the line given by $x^2+2 h x y+6 y^2=0$ are options and the angle between these lines is $\tan ^{-1}\left(\frac{1}{7}\right)$, then the product of the perpendiculars draw from the point $(1,0)$ to the given pair of lines is

If one of the lines represented by $a x^2+2 h x y+b y^2=0$ bisects the angle between the positive coordinates axes, then

If the slope of one of the pair of lines represented by $2 x^2+3 x y+K y^2=0$ is 2 , then the angle between the pair of lines is

If the equation of the pair of straight lines passing through the point $(1,1)$ and perpendicular to the pair of lines $3 x^2+11 x y-4 y^2=0$ is $a x^2+2 h x y+b y^2+2 g x+2 f y+12=0$, then $2(a-h+b-g+f-12)=$

Suppose $P$ and $Q$ lie on $3 x+4 y-4=0$ and $5 x-y-4=0$ respectively. If the mid-point of $P Q$ is $(1,5)$, then the slope of the line passing through $P$ and $Q$ is

The length of intercept of $x+1=0$ between the lines $3 x+2 y=5$ and $3 x+2 y=3$ is

Suppose the slopes $m_1$ and $m_2$ of the lines represented by $a x^2+2 h x y+b y^2=0$ satisfy $3\left(m_1-m_2\right)-7=0$ and $m_1 m_2-2=0$. Then, which of the following is true?

Suppose that the sides passing through the vertex $(\alpha, \beta)$ of a triangle are bisected at right angles by the lines $y^2-8 x y-9 x^2=0$. Then, the centroid of the triangle is

Suppose $P$ and $Q$ are the mid-points of the sides $A B$ and $B C$ of a triangle where $A(1,3), B(3,7)$ and $C(7,15)$ are vertices. Then, the locus of $R$ satisfying $A C^2+Q R^2=P R^2$ is

If the points of intersection of the coordinate axes and $|x+y|=2$ form a rhombus, then its area is

Suppose, in $\triangle A B C, x-y+5=0, x+2 y=0$ are respectively the equations of the perpendicular bisectors of the sides $A B$ and $A C$. If $A$ is $(1,-2)$, the equation of the line joining $B$ and $C$ is

If the pair of straight lines $9 x^2+a x y+4 y^2+6 x+b y-3=0$ represents two parallel lines, then

A line passing through $P(2,3)$ and making an angle of $30^{\circ}$ with the positive direction of $X$-axis meets $x^2-2 x y-y^2=0$ at $A$ and $B$. Then the value of $P A: P B$ is

The least distance from origin to a point on the line $y=x+3$ which lies at a distance of 2 units from $(0,3)$ is

Starting from the point $A(-3,4)$, a moving object touches $2 x+y-7=0$ at $B$ and reaches the point $C(0,1)$. If the object travels along the shortest path, the distance between $A$ and $B$ is

Suppose a triangle is formed by $x+y=10$ and the coordinate axes. Then, the number of points $(x, y)$ where $x$ and $y$ are natural numbers, lying inside the triangle is

If the lines represented by $a x^2+2 h x y+b y^2+2 g x+2 f y+c=0$ intersect on the $X$-axis, which of the following is in general incorrect?

For $\alpha \in\left[0, \frac{\pi}{2}\right]$, the angle between the lines represented by $[x \cos \theta-y] [(\cos \theta+\tan \alpha) x-(1-\cos \theta \tan \alpha) y]=0$ is

The point to which the origin should be shifted in order to eliminate the $x$ and $y$ terms from the equation $9 x^2+4 y^2+10 x+12 y+1=0$ is

If $A(1,3)$ and $C(7,5)$ are two opposite vertices of a square, then find the equation of a side passing through $A$.

$C$ is the centroid of the triangle with vertices $(3,-1),(1,3)$ and $(2,4)$. Let $P$ be the point of intersection of the lines $x+3 y-1=0$ and $3 x-y+1=0$. Then a line which passes through both points $C$ and $P$ would also passes through the point .......

The distance of the point $(1,2)$ from the line $x+y+5=0$ measured along the line parallel to $3 x-y=7$ is equal to

Find the equation of a line which passes through $\left(2 \cos ^3(\theta), 2 \sin ^3(\theta)\right)$ and is perpendicular to the line $x \cos (\theta)-y \sin (\theta)=2 \cos (2 \theta)$.

The value of $p$ for which the equation $x^2+p x y+y^2-5 x-7 y+6=0$ represents a pair of straight lines is

If one of the line represented by $-a x^2+2 h x y+b y^2=0$ passes through $(2,3)$ and the other passes through $(4,5)$, then $a+2 h+b$ equals

If the lines represented by the equation $2 x^2-p x y+2 y^2=0$ are real, then the value of $p$ lies in the interval

When the axes are rotated through an angle 45$^\circ$, the new coordinates of a point P are (1, $-$1). The coordinates of P in the original system are

Find the equation of a straight line passing through $(-5,6)$ and cutting off equal intercepts on the coordinate axes.

Line has slope $m$ and $y$-intercept 4 . The distance between the origin and the line is equal to

The equation of the base of an equilateral triangle is $x+y=2$ and one vertex is $(2,-1)$, then the length of the side of the triangle is

The equation of a straight line which passes through the point $\left(a \cos ^3 \theta, a \sin ^3 \theta\right)$ and perpendicular to $(x \sec \theta+y \operatorname{cosec} \theta)=a$ is

The acute angle between lines $6 x^2+11 x y-10 y^2=0$ is

If the lines, joining the origin to the points of intersection of the curve $2 x^2-2 x y+3 y^2+2 x-y-1=0$ and the line $x+2 y=k$, are at right angles, then $k^2$ equals

The equation of bisector of the angle between the lines represented by $3 x^2-5 x y+4 y^2=0$ is

If the bisectors of the pair of lines $x^2-2 m x y-y^2=0$ is represented by $x^2-2 n x y-y^2=0$, then

If $A(4,7), B(-7,8)$ and $C(1,2)$ are the vertices of $\triangle A B C$, then the equation of perpendicular bisector of the side $A B$ is

The ratio in which the straight line $3 x+4 y=6$ divides the join of the points $(2,-1)$ and $(1,1)$ is

Find the equation of a line passing through the point $(4,3)$, which cuts a triangle of minimum area from the first quadrant.

If the orthocenter of the triangle formed by the lines $2 x+3 y-1=0, x+2 y+1=0$ and $a x+b y-1=0$ lies at origin, then $\frac{1}{a}+\frac{1}{b}$ is equal to

The equation $8 x^2-24 x y+18 y^2-6 x+9 y-5=0$ represents a

Find the angle between the pair of lines represented by the equation $x^2+4 x y+y^2=0$.

If the acute angle between lines $a x^2+2 h x y+b y^2=0$ is $\frac{\pi}{4}$, then $4 h^2$ is equal to

The angle between the lines represented by $\cos \theta(\cos \theta+1) x^2-\left(2 \cos \theta+\sin ^2 \theta\right) x y+(1-\cos \theta) y^2=0$ is

If the axes are rotated through an angle $45 \Upsilon$, the coordinates of the point $(2 \sqrt{2},-3 \sqrt{2})$ in the new system are

the sum of the squares of the intercepts made the line $5x-2y=10$ on the coordinate axes equals

For three consecutive odd integers $a \cdot b$ and $c$, if the variable line $a x+b y+c=0$ always passes through the point $(\alpha, \beta)$, the value of $\alpha^2+\beta^2$ equals

If $2x+3y+4=0$ is the perpendicular bisector of the line segment joining the points A(1, 2) and B($\alpha,\beta$), then the value of $13\alpha+13\beta$ equals

The equation of the pair of straight lines perpendicular to the pair $2 x^2+3 x y+2 y^2+10 x+5 y=0$ and passing though the origin is

If the centroid of the triangle formed by the lines $2 y^2+5 x y-3 x^2=0$ and $x+y=k$ is $\left(\frac{1}{18}, \frac{11}{18}\right)$, then the value of $k$ equals

If $m_1$ and $m_2,\left(m_1>m_2\right)$ are the slopes of the lines represented by $5 x^2-8 x y+3 y^2=0$, then $m_1: m_2$ equals

If the slope of one of the lines represented by $a x^2+2 h x y+b y^2=0$ is the square of the other then, $\left|\frac{a+b}{h}+\frac{8 h^2}{a b}\right|$ is equal to