Coordinate Geometry PYQ — Page 7

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

Let $C$ be the circle of minimum area enclosing the ellipse $E: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with eccentricity $\frac{1}{2}$ and foci $( \pm 2,0)$. Let PQR be a variable triangle, whose vertex $P$ is on the circle $C$ and the side $Q R$ of length 29 is parallel to the major axis of $E$ and contains the point of intersection of $E$ with the negative $y$-axis. Then the maximum area of the triangle PQR is :

Let \(A B C\) be a triangle formed by the lines \(7 x-6 y+3=0, x+2 y-31=0\) and \(9 x-2 y-19=0\). Let the point \((h, k)\) be the image of the centroid of \(\Delta A B C\) in the line \(3 x+6 y-53=0\). Then \(h^2+k^2+h k\) is equal to:

Let P be the parabola, whose focus is $(-2,1)$ and directrix is $2 x+y+2=0$. Then the sum of the ordinates of the points on P , whose abscissa is -2 , is

The axis of a parabola is the line $y=x$ and its vertex and focus are in the first quadrant at distances $\sqrt{2}$ and $2 \sqrt{2}$ units from the origin, respectively. If the point $(1, \mathrm{k})$ lies on the parabola, then a possible value of $k$ is :-

Let the area of the triangle formed by a straight Line L: $\mathrm{x}+\mathrm{by}+\mathrm{c}=0$ with co-ordinate axes be 48 square units. If the perpendicular drawn from the origin to the line L makes an angle of $45^{\circ}$ with the positive x -axis, then the value of $\mathrm{b}^2+\mathrm{c}^2$ is:

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 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 $P$ 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

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

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

If the circles $(x+1{)}^{2}+(y+2{)}^{2}={r}^{2}$ and ${x}^{2}+{y}^{2}-4x-4y+4=0$ intersect at exactly two distinct points, then

Consider a triangle $\mathrm{ABC}$ having the vertices $\mathrm{A}(1,2), \mathrm{B}(\alpha, \beta)$ and $\mathrm{C}(\gamma, \delta)$ and angles $\angle A B C=\frac{\pi}{6}$ and $\angle B A C=\frac{2 \pi}{3}$. If the points $\mathrm{B}$ and $\mathrm{C}$ lie on the line $y=x+4$, then $\alpha^2+\gamma^2$ 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

Let a ray of light passing through the point $(3,10)$ reflects on the line $2 x+y=6$ and the reflected ray passes through the point $(7,2)$. If the equation of the incident ray is $a x+b y+1=0$, then $a^2+b^2+3 a b$ is equal to_________

If the orthocentre of the triangle formed by the lines $2 x+3 y-1=0, x+2 y-1=0$ and $a x+b y-1=0$, is the centroid of another triangle, whose circumcentre and orthocentre respectively are $(3,4)$ and $(-6,-8)$, then the value of $|a-b|$ 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

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 a conic $C$ pass through the point $(4,-2)$ and $P(x, y), x \geq 3$, be any point on $C$. Let the slope of the line touching the conic $C$ only at a single point $P$ be half the slope of the line joining the points $P$ and $(3,-5)$. If the focal distance of the point $(7,1)$ on $C$ is $d$, then $12 d$ equals ______

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?