Coordinate Geometry PYQ — Page 7

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 :

A line passing through the point $\mathrm{A}(-2,0)$, touches the parabola $P: y^2=x-2$ at the point $B$ in the first quadrant. The area, of the region bounded by the line AB , parabola P and the x -axis, is :-

A line passing through the point $\mathrm{P}(\mathrm{a}, \theta)$ 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 clock-wise 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 $3 a^2 \tan ^2 \alpha-2 \sqrt{3}$ is

A line passing through the point $\mathrm{P}(\sqrt{5}, \sqrt{5})$ intersects the ellipse $\frac{\mathrm{x}^2}{36}+\frac{\mathrm{y}^2}{25}=1$ at $A$ and $B$ such that $(P A) .(P B)$ is maximum. Then $5\left(P A^2+P B^2\right)$ is equal to :

A line passes through the origin and makes equal angles with the positive coordinate axes. It intersects the lines $\mathrm{L}_1: 2 \mathrm{x}+\mathrm{y}+6=0$ and $\mathrm{L}_2: 4 \mathrm{x}+2 \mathrm{y}-\mathrm{p}=0, \mathrm{p} \gt 0$, 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

A circle $C$ of radius 2 lies in the second quadrant and touches both the coordinate axes. Let $r$ be the radius of a circle that has centre at the point $(2,5)$ and intersects the circle $C$ at exactly two points. If the set of all possible values of r is the interval $(\alpha, \beta)$, then $3 \beta-2 \alpha$ is equal to :

Two vertices of a triangle $\mathrm{ABC}$ are $\mathrm{A}(3,-1)$ and $\mathrm{B}(-2,3)$, and its orthocentre is $\mathrm{P}(1,1)$. If the coordinates of the point $\mathrm{C}$ are $(\alpha, \beta)$ and the centre of the of the circle circumscribing the triangle $\mathrm{PAB}$ is $(\mathrm{h}, \mathrm{k})$, then the value of $(\alpha+\beta)+2(\mathrm{~h}+\mathrm{k})$ equals

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 :

The portion of the line $4x+5y=20$ in the first quadrant is trisected by the lines ${L}_{1}$ and ${L}_{2}$ passing through the origin. The tangent of an angle between the lines ${L}_{1}$ and ${L}_{2}$ is :

The maximum area of a triangle whose one vertex is at $(0,0)$ and the other two vertices lie on the curve $y=-2{x}^{2}+54$ at points $(x,y)$ and $(-x,y)$ where $y>0$ is :

The length of the chord of the ellipse $\frac{{x}^{2}}{25}+\frac{{y}^{2}}{16}=1$, whose mid point is $(1,\frac{2}{5})$, is equal to:

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

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

Suppose $A B$ is a focal chord of the parabola $y^2=12 x$ of length $l$ and slope $\mathrm{m} < \sqrt{3}$. If the distance of the chord $\mathrm{AB}$ from the origin is $\mathrm{d}$, then $l \mathrm{~d}^2$ is equal to _______

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 :

Let the locus of the mid points of the chords of circle ${x}^{2}+{(y-1)}^{2}=1$ drawn from the origin intersect the line $x+y=1$ at $P$ and $Q$. Then, the length of $PQ$ is:

Let the line $L:\sqrt{2}x+y=\alpha$ pass through the point of the intersection $P$(in the first quadrant)of the circle ${x}^{2}+{y}^{2}=3$ and the parabola ${x}^{2}=2y$. Let the line $L$ touch two circles ${C}_{1}$ and ${C}_{2}$ of equal radius $2\sqrt{3}$. If the centres ${Q}_{1}$ and ${Q}_{2}$ of the circles ${C}_{1}$ and ${C}_{2}$ lie on the $y-$axis, then the square of the area of the triangle $P{Q}_{1}{Q}_{2}$ is equal to _________.

Let the line $2 x+3 y-\mathrm{k}=0, \mathrm{k}>0$, intersect the $x$-axis and $y$-axis at the points $\mathrm{A}$ and $\mathrm{B}$, respectively. If the equation of the circle having the line segment $\mathrm{AB}$ as a diameter is $x^2+y^2-3 x-2 y=0$ and the length of the latus rectum of the ellipse $x^2+9 y^2=k^2$ is $\frac{m}{n}$, where $m$ and $n$ are coprime, then $2 \mathrm{~m}+\mathrm{n}$ is equal to

Let the length of the focal chord PQ of the parabola $y^2=12 x$ be 15 units. If the distance of $\mathrm{PQ}$ from the origin is $\mathrm{p}$, then $10 \mathrm{p}^2$ is equal to _______

Let the latus rectum of the hyperbola $\frac{{x}^{2}}{9}-\frac{{y}^{2}}{{b}^{2}}=1$ subtend an angle of $\frac{\pi }{3}$ at the centre of the hyperbola. If ${b}^{2}$ is equal to $\frac{l}{m}(1+\sqrt{n})$, where $l$ and $m$ are co-prime numbers, then ${l}^{2}+{m}^{2}+{n}^{2}$ is equal to __________.

Let the foci of a hyperbola $H$ coincide with the foci of the ellipse $E: \frac{(x-1)^2}{100}+\frac{(y-1)^2}{75}=1$ and the eccentricity of the hyperbola $H$ be the reciprocal of the eccentricity of the ellipse $E$. If the length of the transverse axis of $H$ is $\alpha$ and the length of its conjugate axis is $\beta$, then $3 \alpha^2+2 \beta^2$ is equal to

Let the foci and length of the latus rectum of an ellipse $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1$, $a>b$ be $(\pm 5,0)$ and $\sqrt{50}$, respectively. Then, the square of the eccentricity of the hyperbola $\frac{{x}^{2}}{{b}^{2}}-\frac{{y}^{2}}{{a}^{2}{b}^{2}}=1$ equals

Let the circles $C_1:(x-\alpha)^2+(y-\beta)^2=r_1^2$ and $C_2:(x-8)^2+\left(y-\frac{15}{2}\right)^2=r_2^2$ touch each other externally at the point $(6,6)$. If the point $(6,6)$ divides the line segment joining the centres of the circles $C_1$ and $C_2$ internally in the ratio $2: 1$, then $(\alpha+\beta)+4\left(r_1^2+r_2^2\right)$ equals

Let the circle $C_1: x^2+y^2-2(x+y)+1=0$ and $C_2$ be a circle having centre at $(-1,0)$ and radius 2 . If the line of the common chord of $\mathrm{C}_1$ and $\mathrm{C}_2$ intersects the $y$-axis at the point $\mathrm{P}$, then the square of the distance of $\mathrm{P}$ from the centre of $\mathrm{C}_1$ is :