Let the angles made with the positive $x$-axis by two straight lines drawn from the point $\mathrm{P}(2,3)$ and meeting the line $x+y=6$ at a distance $\sqrt{\frac{2}{3}}$ from the point P be $\theta_1$ and $\theta_2$. Then the value of $\left(\theta_1+\theta_2\right)$ is:

Let $A(1,0), B(2,-1)$ and $C\left(\frac{7}{3}, \frac{4}{3}\right)$ be three points. If the equation of the bisector of the angle ABC is $\alpha x+\beta y=5$, then the value of $\alpha^2+\beta^2$ is

Let $\mathrm{A}(1,2)$ and $\mathrm{C}(-3,-6)$ be two diagonally opposite vertices of a rhombus, whose sides AD and BC are parallel to the line $7 x-y=14$. If $\mathrm{B}(\alpha, \beta)$ and $\mathrm{D}(\gamma, \delta)$ are the other two vertices, then $|\alpha+\beta+\gamma+\delta|$ is equal to :

A rectangle is formed by the lines $x=0, y=0, x=3$ and $y=4$. Let the line L be perpendicular to $3 x+y+6=0$ and divide the area of the rectangle into two equal parts. Then the distance of the point $\left(\frac{1}{2},-5\right)$ from the line $L$ is equal to :

Among the statements

$(S 1)$ : If $A(5,-1)$ and $B(-2,3)$ are two vertices of a triangle, whose orthocentre is $(0,0)$, then its third vertex is $(-4,-7)$

and

(S2) : If positive numbers $2 a, b, c$ are three consecutive terms of an A.P., then the lines $a x+b y+c=0$ are concurrent at $(2,-2)$,

Let a point A lie between the parallel lines $\mathrm{L}_1$ and $\mathrm{L}_2$ such that its distances from $\mathrm{L}_1$ and $\mathrm{L}_2$ are 6 and 3 units, respectively. Then the area (in sq. units) of the equilateral triangle ABC , where the points B and C lie on the lines $\mathrm{L}_1$ and $\mathrm{L}_2$, respectively, is :

If a straight line drawn through the point of intersection of the lines $4 x+3 y-1=0$ and $3 x+4 y-1=0$, meets the co-ordinate axes at the points P and Q , then the locus of the mid point of PQ is :

In an equilateral triangle $P Q R$, let the vertex $P$ be at $(3,5)$ and the side $Q R$ be along the line $x+y=4$. If the orthocentre of the triangle PQR is $(\alpha, \beta)$, then $9(\alpha+\beta)$ is equal to:

Let the line $\mathrm{L}_1: x+3=0$ intersect the lines $\mathrm{L}_2: x-y=0$ and $\mathrm{L}_3: 3 x+y=0$ at the points A and B , respectively. Let the bisector of the obtuse angle between the lines $L_2$ and $L_3$ intersect the line $L_1$ at the point $C$. Then $B C^2: A C^2$ is equal to:

Let the vertex A of a triangle ABC be $(1,2)$, and the mid-point of the side AB be $(5,-1)$. If the centroid of this triangle is $(3,4)$ and its circumcenter is $(\alpha, \beta)$, then $21(\alpha+\beta)$ is equal to :

Let the mid points of the sides of a triangle ABC be $\left(\frac{5}{2}, 7\right)$, $\left(\frac{5}{2}, 3\right)$ and $(4, 5)$. If its incentre is $(h, k)$, then $3h + k$ is equal to :

Let a be the length of a side of a square OABC with O being the origin. Its side OA makes an acute angle $\alpha $ with the positive x-axis and the equations of its diagonals are $(\sqrt{3}+1)x+(\sqrt{3}-1)y=0$ and $(\sqrt{3}-1)x-(\sqrt{3}+1)y+8\sqrt{3}=0$. Then $a$² is equal to :

A line passing through the point P($a$, 0) makes an acute angle $\alpha $ with the positive x-axis. Let this line be rotated about the point P through an angle $\frac{\alpha}{2}$ in the clockwise direction. If in the new position, the slope of the line is $2 - \sqrt{3}$ and its distance from the origin is $\frac{1}{\sqrt{2}}$, then the value of $3a^2 \tan^2 \alpha - 2\sqrt{3}$ is :

If the orthocenter of the triangle formed by the lines y = x + 1, y = 4x - 8 and y = mx + c is at (3, -1), then m - c is :

Let ABC be the triangle such that the equations of lines AB and AC be $3 y-x=2$ and $x+y=2$, respectively, and the points B and C lie on $x$-axis. If P is the orthocentre of the triangle ABC , then the area of the triangle PBC is equal to

Let the three sides of a triangle are on the lines $4 x-7 y+10=0, x+y=5$ and $7 x+4 y=15$. Then the distance of its orthocentre from the orthocentre of the tringle formed by the lines $x=0, y=0$ and $x+y=1$ is

A line passes through the origin and makes equal angles with the positive coordinate axes. It intersects the lines $\mathrm{L}_1: 2 x+y+6=0$ and $\mathrm{L}_2: 4 x+2 y-p=0, p>0$, at the points A and B , respectively. If $A B=\frac{9}{\sqrt{2}}$ and the foot of the perpendicular from the point $A$ on the line $L_2$ is $M$, then $\frac{A M}{B M}$ is equal to

Let the line x + y = 1 meet the axes of x and y at A and B, respectively. A right angled triangle AMN is inscribed in the triangle OAB, where O is the origin and the points M and N lie on the lines OB and AB, respectively. If the area of the triangle AMN is $ \frac{4}{9} $ of the area of the triangle OAB and AN : NB = $ \lambda : 1 $, then the sum of all possible value(s) of $ \lambda $ is:

Let ΔABC be a triangle formed by the lines 7x – 6y + 3 = 0, x + 2y – 31 = 0 and 9x – 2y – 19 = 0. Let the point (h, k) be the image of the centroid of ΔABC in the line 3x + 6y – 53 = 0. Then h² + k² + hk is equal to :

Two equal sides of an isosceles triangle are along $ -x + 2y = 4 $ and $ x + y = 4 $. If $ m $ is the slope of its third side, then the sum, of all possible distinct values of $ m $, is:

If A and B are the points of intersection of the circle $x^2 + y^2 - 8x = 0$ and the hyperbola $\frac{x^2}{9} - \frac{y^2}{4} = 1$ and a point P moves on the line $2x - 3y + 4 = 0$, then the centroid of $\Delta PAB$ lies on the line :

Let the points $\left(\frac{11}{2}, \alpha\right)$ lie on or inside the triangle with sides $x+y=11, x+2 y=16$ and $2 x+3 y=29$. Then the product of the smallest and the largest values of $\alpha$ is equal to :

Let the lines $3 x-4 y-\alpha=0,8 x-11 y-33=0$, and $2 x-3 y+\lambda=0$ be concurrent. If the image of the point $(1,2)$ in the line $2 x-3 y+\lambda=0$ is $\left(\frac{57}{13}, \frac{-40}{13}\right)$, then $|\alpha \lambda|$ is equal to

A rod of length eight units moves such that its ends $A$ and $B$ always lie on the lines $x-y+2=0$ and $y+2=0$, respectively. If the locus of the point $P$, that divides the rod $A B$ internally in the ratio $2: 1$ is $9\left(x^2+\alpha y^2+\beta x y+\gamma x+28 y\right)-76=0$, then $\alpha-\beta-\gamma$ is equal to :

Let the triangle PQR be the image of the triangle with vertices $(1,3),(3,1)$ and $(2,4)$ in the line $x+2 y=2$. If the centroid of $\triangle \mathrm{PQR}$ is the point $(\alpha, \beta)$, then $15(\alpha-\beta)$ is equal to :

A variable line $\mathrm{L}$ passes through the point $(3,5)$ and intersects the positive coordinate axes at the points $\mathrm{A}$ and $\mathrm{B}$. The minimum area of the triangle $\mathrm{OAB}$, where $\mathrm{O}$ is the origin, is :

A ray of light coming from the point $\mathrm{P}(1,2)$ gets reflected from the point $\mathrm{Q}$ on the $x$-axis and then passes through the point $R(4,3)$. If the point $S(h, k)$ is such that $P Q R S$ is a parallelogram, then $hk^2$ is equal to:

If the line segment joining the points $(5,2)$ and $(2, a)$ subtends an angle $\frac{\pi}{4}$ at the origin, then the absolute value of the product of all possible values of $a$ is :

The equations of two sides $\mathrm{AB}$ and $\mathrm{AC}$ of a triangle $\mathrm{ABC}$ are $4 x+y=14$ and $3 x-2 y=5$, respectively. The point $\left(2,-\frac{4}{3}\right)$ divides the third side $\mathrm{BC}$ internally in the ratio $2: 1$, the equation of the side $\mathrm{BC}$ is

If the locus of the point, whose distances from the point $(2,1)$ and $(1,3)$ are in the ratio $5: 4$, is $a x^2+b y^2+c x y+d x+e y+170=0$, then the value of $a^2+2 b+3 c+4 d+e$ is equal to :

Let a variable line of slope $m>0$ passing through the point $(4,-9)$ intersect the coordinate axes at the points $A$ and $B$. The minimum value of the sum of the distances of $A$ and $B$ from the origin is

Let $\mathrm{A}(-1,1)$ and $\mathrm{B}(2,3)$ be two points and $\mathrm{P}$ be a variable point above the line $\mathrm{AB}$ such that the area of $\triangle \mathrm{PAB}$ is 10. If the locus of $\mathrm{P}$ is $\mathrm{a} x+\mathrm{by}=15$, then $5 \mathrm{a}+2 \mathrm{~b}$ is :

Let two straight lines drawn from the origin $\mathrm{O}$ intersect the line $3 x+4 y=12$ at the points $\mathrm{P}$ and $\mathrm{Q}$ such that $\triangle \mathrm{OPQ}$ is an isosceles triangle and $\angle \mathrm{POQ}=90^{\circ}$. If $l=\mathrm{OP}^2+\mathrm{PQ}^2+\mathrm{QO}^2$, then the greatest integer less than or equal to $l$ is :

The vertices of a triangle are $\mathrm{A}(-1,3), \mathrm{B}(-2,2)$ and $\mathrm{C}(3,-1)$. A new triangle is formed by shifting the sides of the triangle by one unit inwards. Then the equation of the side of the new triangle nearest to origin is :

Let $A(a, b), B(3,4)$ and $C(-6,-8)$ respectively denote the centroid, circumcentre and orthocentre of a triangle. Then, the distance of the point $P(2 a+3,7 b+5)$ from the line $2 x+3 y-4=0$ measured parallel to the line $x-2 y-1=0$ is

Let $\alpha, \beta, \gamma, \delta \in \mathbb{Z}$ and let $A(\alpha, \beta), B(1,0), C(\gamma, \delta)$ and $D(1,2)$ be the vertices of a parallelogram $\mathrm{ABCD}$. If $A B=\sqrt{10}$ and the points $\mathrm{A}$ and $\mathrm{C}$ lie on the line $3 y=2 x+1$, then $2(\alpha+\beta+\gamma+\delta)$ is equal to

If $x^2-y^2+2 h x y+2 g x+2 f y+c=0$ is the locus of a point, which moves such that it is always equidistant from the lines $x+2 y+7=0$ and $2 x-y+8=0$, then the value of $g+c+h-f$ equals

A line passing through the point $\mathrm{A}(9,0)$ makes an angle of $30^{\circ}$ with the positive direction of $x$-axis. If this line is rotated about A through an angle of $15^{\circ}$ in the clockwise direction, then its equation in the new position is :

Let $\mathrm{A}$ be the point of intersection of the lines $3 x+2 y=14,5 x-y=6$ and $\mathrm{B}$ be the point of intersection of the lines $4 x+3 y=8,6 x+y=5$. The distance of the point $P(5,-2)$ from the line $\mathrm{AB}$ is

The distance of the point $(2,3)$ from the line $2 x-3 y+28=0$, measured parallel to the line $\sqrt{3} x-y+1=0$, is equal to

In a $\triangle A B C$, suppose $y=x$ is the equation of the bisector of the angle $B$ and the equation of the side $A C$ is $2 x-y=2$. If $2 A B=B C$ and the points $A$ and $B$ are respectively $(4,6)$ and $(\alpha, \beta)$, then $\alpha+2 \beta$ is equal to

Let $\mathrm{R}$ be the interior region between the lines $3 x-y+1=0$ and $x+2 y-5=0$ containing the origin. The set of all values of $a$, for which the points $\left(a^2, a+1\right)$ lie in $R$, is :

Let $(\alpha, \beta)$ be the centroid of the triangle formed by the lines $15 x-y=82,6 x-5 y=-4$ and $9 x+4 y=17$. Then $\alpha+2 \beta$ and $2 \alpha-\beta$ are the roots of the equation :

If the point $\left(\alpha, \frac{7 \sqrt{3}}{3}\right)$ lies on the curve traced by the mid-points of the line segments of the lines $x \cos \theta+y \sin \theta=7, \theta \in\left(0, \frac{\pi}{2}\right)$ between the co-ordinates axes, then $\alpha$ is equal to :

Let $C(\alpha, \beta)$ be the circumcenter of the triangle formed by the lines

$4 x+3 y=69$

$4 y-3 x=17$, and

$x+7 y=61$.

Then $(\alpha-\beta)^{2}+\alpha+\beta$ is equal to :

The straight lines $\mathrm{l_{1}}$ and $\mathrm{l_{2}}$ pass through the origin and trisect the line segment of the line L : $9 x+5 y=45$ between the axes. If $\mathrm{m}_{1}$ and $\mathrm{m}_{2}$ are the slopes of the lines $\mathrm{l_{1}}$ and $\mathrm{l_{2}}$, then the point of intersection of the line $\mathrm{y=\left(m_{1}+m_{2}\right)}x$ with L lies on :

The combined equation of the two lines $ax+by+c=0$ and $a'x+b'y+c'=0$ can be written as

$(ax+by+c)(a'x+b'y+c')=0$.

The equation of the angle bisectors of the lines represented by the equation $2x^2+xy-3y^2=0$ is :

If the orthocentre of the triangle, whose vertices are (1, 2), (2, 3) and (3, 1) is $(\alpha,\beta)$, then the quadratic equation whose roots are $\alpha+4\beta$ and $4\alpha+\beta$, is :

Let $B$ and $C$ be the two points on the line $y+x=0$ such that $B$ and $C$ are symmetric with respect to the origin. Suppose $A$ is a point on $y-2 x=2$ such that $\triangle A B C$ is an equilateral triangle. Then, the area of the $\triangle A B C$ is :

A light ray emits from the origin making an angle 30$^\circ$ with the positive $x$-axis. After getting reflected by the line $x+y=1$, if this ray intersects $x$-axis at Q, then the abscissa of Q is :

Let $m_{1}, m_{2}$ be the slopes of two adjacent sides of a square of side a such that $a^{2}+11 a+3\left(m_{1}^{2}+m_{2}^{2}\right)=220$. If one vertex of the square is $(10(\cos \alpha-\sin \alpha), 10(\sin \alpha+\cos \alpha))$, where $\alpha \in\left(0, \frac{\pi}{2}\right)$ and the equation of one diagonal is $(\cos \alpha-\sin \alpha) x+(\sin \alpha+\cos \alpha) y=10$, then $72\left(\sin ^{4} \alpha+\cos ^{4} \alpha\right)+a^{2}-3 a+13$ is equal to :

Let $\mathrm{A}(\alpha,-2), \mathrm{B}(\alpha, 6)$ and $\mathrm{C}\left(\frac{\alpha}{4},-2\right)$ be vertices of a $\triangle \mathrm{ABC}$. If $\left(5, \frac{\alpha}{4}\right)$ is the circumcentre of $\triangle \mathrm{ABC}$, then which of the following is NOT correct about $\triangle \mathrm{ABC}$?

Let the circumcentre of a triangle with vertices A(a, 3), B(b, 5) and C(a, b), ab > 0 be P(1,1). If the line AP intersects the line BC at the point Q$\left(k_{1}, k_{2}\right)$, then $k_{1}+k_{2}$ is equal to :

The equations of the sides $\mathrm{AB}, \mathrm{BC}$ and CA of a triangle ABC are $2 x+y=0, x+\mathrm{p} y=39$ and $x-y=3$ respectively and $\mathrm{P}(2,3)$ is its circumcentre. Then which of the following is NOT true?

Let $A(1,1), B(-4,3), C(-2,-5)$ be vertices of a triangle $A B C, P$ be a point on side $B C$, and $\Delta_{1}$ and $\Delta_{2}$ be the areas of triangles $A P B$ and $A B C$, respectively. If $\Delta_{1}: \Delta_{2}=4: 7$, then the area enclosed by the lines $A P, A C$ and the $x$-axis is :

A point $P$ moves so that the sum of squares of its distances from the points $(1,2)$ and $(-2,1)$ is 14. Let $f(x, y)=0$ be the locus of $\mathrm{P}$, which intersects the $x$-axis at the points $\mathrm{A}$, $\mathrm{B}$ and the $y$-axis at the points C, D. Then the area of the quadrilateral ACBD is equal to :

Let the point $P(\alpha, \beta)$ be at a unit distance from each of the two lines $L_{1}: 3 x-4 y+12=0$, and $L_{2}: 8 x+6 y+11=0$. If $P$ lies below $L_{1}$ and above ${ }{L_{2}}$, then $100(\alpha+\beta)$ is equal to :

A line, with the slope greater than one, passes through the point $A(4,3)$ and intersects the line $x-y-2=0$ at the point B. If the length of the line segment $A B$ is $\frac{\sqrt{29}}{3}$, then $B$ also lies on the line :

Let $\alpha$₁, $\alpha$₂ ($\alpha$₁ < $\alpha$₂) be the values of $\alpha$ fo the points ($\alpha$, $-$3), (2, 0) and (1, $\alpha$) to be collinear. Then the equation of the line, passing through ($\alpha$₁, $\alpha$₂) and making an angle of ${\pi \over 3}$ with the positive direction of the x-axis, is :

The distance of the origin from the centroid of the triangle whose two sides have the equations $x - 2y + 1 = 0$ and $2x - y - 1 = 0$ and whose orthocenter is $\left( {{7 \over 3},{7 \over 3}} \right)$ is :

The distance between the two points A and A' which lie on y = 2 such that both the line segments AB and A' B (where B is the point (2, 3)) subtend angle ${\pi \over 4}$ at the origin, is equal to :

Let a triangle be bounded by the lines L₁ : 2x + 5y = 10; L₂ : $-$4x + 3y = 12 and the line L₃, which passes through the point P(2, 3), intersects L₂ at A and L₁ at B. If the point P divides the line-segment AB, internally in the ratio 1 : 3, then the area of the triangle is equal to :

In an isosceles triangle ABC, the vertex A is (6, 1) and the equation of the base BC is 2x + y = 4. Let the point B lie on the line x + 3y = 7. If ($\alpha$, $\beta$) is the centroid of $\Delta$ABC, then 15($\alpha$ + $\beta$) is equal to :

Let R be the point (3, 7) and let P and Q be two points on the line x + y = 5 such that PQR is an equilateral triangle. Then the area of $\Delta$PQR is :

Let the area of the triangle with vertices A(1, $\alpha$), B($\alpha$, 0) and C(0, $\alpha$) be 4 sq. units. If the points ($\alpha$, $-$$\alpha$), ($-$$\alpha$, $\alpha$) and ($\alpha$², $\beta$) are collinear, then $\beta$ is equal to :

Statement-1: The ratio $PR$ : $RQ$ equals $2\sqrt 2 :\sqrt 5 $
Statement-2: In any triangle, bisector of an angle divide the triangle into two similar triangles.