Coordinate Geometry PYQ — Page 21

The tangent to the circle $C_1: x^2+y^2-2 x-1=0$ at the point $(2,1)$ cuts off a chord of length 4 from a circle $C_2$ whose centre is $(3,-2)$. The radius of $C_2$ is

The locus of the point of intersection of the straight lines, $tx-2y-3t=0$ and $x-2ty+3=0 (t\in R),$ is:

If a point $P(0,-2)$ and $Q$ is any point on the circle, ${x}^{2}+{y}^{2}-5x-y+5=0$, then the maximum value of ${(PQ)}^{2}$ is

Let $k$ be an integer such that the triangle with vertices $(k,-3k), (5,k)$ and $(-k, 2)$ has area $28$ sq. units. Then the orthocenter of this triangle is at the point:

A square, of each side $2$, lies above the $x$-axis and has one vertex at the origin. If one of the sides passing through the origin makes an angle $30^{\circ}$ with the positive direction of the $x$-axis , then the sum of the $x$-coordinates of the vertices of the square is$:$

Consider an ellipse, whose center is at the origin and its major axis is along the $x$-axis. If its eccentricity is $\frac{3}{5}$ and the distance between its foci is $6$, then the area (in sq. units) of the quadrilateral inscribed in the ellipse, with the vertices as the vertices of the ellipse, is:

The eccentricity of an ellipse whose centre is at the origin is $\frac{1}{2}$ . If one of its directrices is $x=-4$ , then the equation of the normal to it at $(1,\frac{3}{2})$ is:

A line drawn through the point $P(4,7)$ cuts the circle ${x}^{2}+{y}^{2}=9$ at the points $A$ and $B$. Then ${P}_{A}\cdot {P}_{B}$ is equal to.

The eccentricity of an ellipse having centre at the origin, axes along the co-ordinate axes and passing through the points $(4,-1)$ and $(-2,2)$ is

The radius of a circle, having minimum area, which touches the curve $y=4-{x}^{2}$ and the lines, $y=|x|$ is:

If two parallel chords of a circle, having diameter $4$ units, lie on the opposite sides of the center and subtend angles ${\mathrm{cos}}^{-1}⁡(\frac{1}{7})$ and ${\mathrm{sec}}^{-1}⁡(7)$ at the center respectively, then the distance between these chords is:

A straight line through origin $O$ meets the lines $3y=10-4x$and $8x+6y+5=0$ at points $A$ and $B$ respectively. Then, $O$ divides the segment $AB$ in the ratio

If a variable line drawn through the intersection of the lines $\frac{x}{3}+\frac{y}{4}=1$ and $\frac{x}{4}+\frac{y}{3}=1$ , meets the coordinate axes at $A$ and $B$, $(A \neq B),$then the locus of the midpoint of $\mathrm{AB}$ is:

The eccentricity of the hyperbola whose length of its conjugate axis is equal to half of the distance between its foci, is

Equation of the tangent to the circle, at the point $(1,-1)$, whose center, is the point of intersection of the straight lines $x-y=1$ and $2x+y=3$ is:

If one of the diameters of the circle, given by the equation, ${x}^{2}+{y}^{2}-4x+6y-12=0,$ is a chord of a circle $S$, whose centre is at $(-3,2)$, then the radius of $S$ is

Two sides of a rhombus are along the lines, $x-y+1=0$ and $7x-y-5=0$ . If its diagonals intersect at $(-1, -2)$ , then which one of the following is a vertex of this rhombus ?

Let $a$ and $b$ respectively be the semi-transverse and semi-conjugate axes of a standard hyperbola whose eccentricity satisfies the equation $9{e}^{2}-18e+5=0$. If $S(5, 0)$ is a focus and $5x=9$ is the corresponding directrix of this hyperbola, then ${a}^{2}-{b}^{2}$ is equal to

The point $(2,1)$ is translated parallel to the line $L:x-y=4$ by $2\sqrt{3}$ units. If the new point $Q$ lies in the third quadrant, then the equation of the line passing through $Q$ and perpendicular to $L$ is

A hyperbola whose transverse axis is along the major axis of the conic $\frac{{x}^{2}}{3}+\frac{{y}^{2}}{4}=4$ and has vertices at the foci of the conic. If the eccentricity of the hyperbola is $\frac{3}{2}$, then which of the following points does not lie on the hyperbola $?$

A ray of light is incident along a line which meets another line $7x-y+1=0$ at the point $(0,1)$. The ray is then reflected from this point along the line $y+2x=1$. Then the equation of the line of incidence of the ray of light is :

A circle passes through $(-2,4)$ and touches the $y-$axis at $(0,2)$. Which one of the following equations can represent a diameter of this circle ?

Let the tangents drawn to the circle, ${x}^{2}+{y}^{2}=16$ from the point$P(0,h)$ meet the $x$-axis at points $A$ and $B$. If the area of $\Delta APB$ is minimum, then positive value of $h$ is:

If the incentre of an equilateral triangle is $(1,1)$ and the equation of its one side is $3x+4y+3=0$, then the equation of the circumcircle of this triangle is: