Coordinate Geometry PYQ — Page 7

Let the shortest distance from $(\mathrm{a}, 0), \mathrm{a}\gt0$, to the parabola $y^2=4 x$ be 4 . Then the equation of the circle passing through the point $(a, 0)$ and the focus of the parabola, and having its centre on the axis of the parabola is :

If the line $3 x-2 y+12=0$ intersects the parabola $4 y=3 x^2$ at the points $A$ and $B$, then at the vertex of the parabola, the line segment $A B$ subtends an angle 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 $O B$ and $A B$, respectively. If the area of the triangle $A M N$ is $\frac{4}{9}$ of the area of the triangle $O A B$ and AN : NB $=\lambda: 1$, then the sum of all possible value(s) of is $\lambda$ :

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 $\mathrm{a}^2$ is equal to

Let the equation $\mathrm{x}(\mathrm{x}+2)(12-\mathrm{k})=2$ have equal roots. Then the distance of the point $\left(\mathrm{k}, \frac{\mathrm{k}}{2}\right)$ from the line $3 x+4 y+5=0$ is

Let the arc $A C$ of a circle subtend a right angle at the centre $O$. If the point $B$ on the arc $A C$, divides the arc $A C$ such that $\frac{\text { length of } \operatorname{arc} A B}{\text { length of } \operatorname{arc} B C}=\frac{1}{5}$, and $\overrightarrow{O C}=\alpha \overrightarrow{O A}+\beta \overrightarrow{O B}$, then $\alpha+\sqrt{2}(\sqrt{3}-1) \beta$ is equal to

If $\mathrm{P}(6,1)$ be the orthocentre of the triangle whose vertices are $\mathrm{A}(5,-2), \mathrm{B}(8,3)$ and $\mathrm{C}(\mathrm{h}, \mathrm{k})$, then the point $C$ lies on the circle:

Let $P$ be a parabola with vertex $(2,3)$ and directrix $2x+y=6$. Let an ellipse $E:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1,a>b$ of eccentricity $\frac{1}{\sqrt{2}}$ pass through the focus of the parabola $P$. Then the square of the length of the latus rectum of $E$, is

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

Let $A$ be a square matrix of order 2 such that $|A|=2$ and the sum of its diagonal elements is -3 . If the points $(x, y)$ satisfying $\mathrm{A}^2+x \mathrm{~A}+y \mathrm{I}=\mathrm{O}$ lie on a hyperbola, whose $\text { length of semi major axis is } x \text { and semi minor axis is } y$, eccentricity is $\mathrm{e}$ and the length of the latus rectum is $l$, then $81\left(e^ 4+l^2\right)$ is equal to

A circle is inscribed in an equilateral triangle of side of length 12 . If the area and perimeter of any square inscribed in this circle are $m$ and $n$, respectively, then $m+n^2$ is equal to

Consider a hyperbola $\mathrm{H}$ having centre at the origin and foci on the $\mathrm{x}$-axis. Let $\mathrm{C}_1$ be the circle touching the hyperbola $\mathrm{H}$ and having the centre at the origin. Let $\mathrm{C}_2$ be the circle touching the hyperbola $\mathrm{H}$ at its vertex and having the centre at one of its foci. If areas (in sq units) of $C_1$ and $C_2$ are $36 \pi$ and $4 \pi$, respectively, then the length (in units) of latus rectum of $\mathrm{H}$ is

If one of the diameters of the circle ${x}^{2}+{y}^{2}-10x+4y+13=0$ is a chord of another circle $C$, whose center is the point of intersection of the lines $2x+3y=12$ and $3x-2y=5$, then the radius of the circle $C$ is

A line passing through the point $A(9,0)$ 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 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 _________.

For $0<\theta <\pi /2$, if the eccentricity of the hyperbola ${x}^{2}-{y}^{2}{\mathrm{cosec}}^{2}\theta =5$ is $\sqrt{7}$ times eccentricity of the ellipse ${x}^{2}{\mathrm{cosec}}^{2}\theta +{y}^{2}=5$, then the value of $\theta$ is:

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

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 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 :

A square is inscribed in the circle $x^2+y^2-10 x-6 y+30=0$. One side of this square is parallel to $y=x+3$. If $\left(x_i, y_i\right)$ are the vertices of the square, then $\mathbf{\Sigma}\left(x_i^2+y_i^2\right)$ is equal to:

Let a circle $C$ of radius 1 and closer to the origin be such that the lines passing through the point $(3,2)$ and parallel to the coordinate axes touch it. Then the shortest distance of the circle $\mathrm{C}$ from the point $(5,5)$ is :

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 PQ be a chord of the parabola $y^2=12 x$ and the midpoint of PQ be at $(4,1)$. Then, which of the following point lies on the line passing through the points $\mathrm{P}$ and Q?

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 :