Tangents and normals

Let $P$ be the point on the parabola $y=x^2$ such that the slope of the tangent to the parabola at the point $P$ is $4$. Let $Q$ be the point in the first quadrant lying on the circle $x^2+y^2=2$ such that the slope of the tangent to the circle at the point $Q$ is $-1$. Let $R$ be the point in the first quadrant lying on the ellipse $x^2+4y^2=8$ such that the slope of the tangent to the ellipse at the point $R$ is $-\frac12$. Then the radius of the circle passing through the points $P,Q$ and $R$ is

Let A be the point $(3,0)$ and circles with variable diameter AB touch the circle $x^2+y^2=36$ internally. Let the curve C be the locus of the point B. If the eccentricity of C is $e$, then $72e^2$ is equal to ________.

Let the straight line $y = 2x$ touch a circle with center $(0, \alpha)$, $\alpha > 0$, and radius $r$ at a point $A_1$. Let $B_1$ be the point on the circle such that the line segment $A_1B_1$ is a diameter of the circle. Let $\alpha + r = 5 + \sqrt{5}$.

Match each entry in List-I to the correct entry in List-II.

The correct option is

Let $C_1$ be the circle of radius 1 with center at the origin. Let $C_2$ be the circle of radius $r$ with center at the point $A = (4,1)$, where $1 < r < 3$. Two distinct common tangents $PQ$ and $ST$ of $C_1$ and $C_2$ are drawn. The tangent $PQ$ touches $C_1$ at $P$ and $C_2$ at $Q$. The tangent $ST$ touches $C_1$ at $S$ and $C_2$ at $T$. Mid points of the line segments $PQ$ and $ST$ are joined to form a line which meets the $x$-axis at a point $B$. If $AB = \sqrt5$, then the value of $r^2$ is _____.

Let $ABC$ be the triangle with $AB = 1$, $AC = 3$ and $\angle BAC = \dfrac\pi2$. If a circle of radius $r > 0$ touches the sides $AB$, $AC$ and also touches internally the circumcircle of the triangle $ABC$, then the value of $r$ is _____.

If $y=m_1x+c_1$ and $y=m_2x+c_2$, $m_1\ne m_2$ are two common tangents of circle $x^2+y^2=2$ and parabola $y^2=x$, then the value of $8|m_1m_2|$ is equal to :

Let the common tangents to the curves $4\left(x^2+y^2\right)=9$ and $y^2=4x$ intersect at the point Q. Let an ellipse, centered at the origin O, has lengths of semi-minor and semi-major axes equal to OQ and 6, respectively. If $e$ and $l$ respectively denote the eccentricity and the length of the latus rectum of this ellipse, then $\dfrac l{e^2}$ is equal to ______.

If $m$ is the slope of a common tangent to the curves $\dfrac{x^2}{16}+\dfrac{y^2}9=1$ and $x^2+y^2=12$, then $12m^2$ is equal to :

Let a circle C of radius 5 lie below the $x$-axis. The line $L_1:4x+3y+2=0$ passes through the centre P of the circle C and intersects the line $L_2:3x-4y-11=0$ at Q. The line $L_2$ touches C at the point Q. Then the distance of P from the line $5x-12y+51=0$ is ______.

If the tangents drawn at the points $O(0,0)$ and $P\left(1+\sqrt5,2\right)$ on the circle $x^2+y^2-2x-4y=0$ intersect at the point Q, then the area of the triangle OPQ is equal to

Let the lines $y+2x=\sqrt{11}+7\sqrt7$ and $2y+x=2\sqrt{11}+6\sqrt7$ be normal to a circle $C:(x-h)^2+(y-k)^2=r^2$. If the line $\sqrt{11}y-3x=\dfrac{5\sqrt{77}}3+11$ is tangent to the circle C, then the value of $(5h-8k)^2+5r^2$ is equal to ______.

Let the tangent to the circle $C_1:x^2+y^2=2$ at the point $M(-1,1)$ intersect the circle $C_2:(x-3)^2+(y-2)^2=5$, at two distinct points A and B. If the tangents to $C_2$ at the points A and B intersect at N, then the area of the triangle ANB is equal to :

Consider three circles :
$$C_1:x^2+y^2=r^2$$
$$C_2:(x-1)^2+(y-1)^2=r^2$$
$$C_3:(x-2)^2+(y-1)^2=r^2$$
If a line $L:y=mx+c$ be a common tangent to $C_1$, $C_2$ and $C_3$ such that $C_1$ and $C_3$ lie on one side of line L while $C_2$ lies on other side, then the value of $20\left(r^2+c\right)$ is equal to :

Let the locus of the centre $(\alpha,\beta)$, $\beta>0$, of the circle which touches the circle $x^2+(y-1)^2=1$ externally and also touches the $x$-axis be L. Then the area bounded by L and the line $y=4$ is :

The sum of diameters of the circles that touch (i) the parabola $75x^2=64(5y-3)$ at the point $\left(\dfrac85,\dfrac65\right)$ and (ii) the $y$-axis, is equal to ______.

A circle $C_1$ passes through the origin O and has diameter 4 on the positive $x$-axis. The line $y=2x$ gives a chord OA of circle $C_1$. Let $C_2$ be the circle with OA as a diameter. If the tangent to $C_2$ at the point A meets the $x$-axis at P and $y$-axis at Q, then QA : AP is equal to :

Let the tangents at two points A and B on the circle $x^2+y^2-4x+3=0$ meet at origin O (0, 0). Then the area of the triangle OAB is :

Two tangent lines $l_1$ and $l_2$ are drawn from the point (2, 0) to the parabola $2y^2=-x$. If the lines $l_1$ and $l_2$ are also tangent to the circle $(x-5)^2+y^2=r$, then 17r is equal to ______.

Let a line L pass through the point of intersection of the lines $bx+10y-8=0$ and $2x-3y=0$, $b\in\mathbf R-\left\{\frac43\right\}$. If the line L also passes through the point (1, 1) and touches the circle $17(x^2+y^2)=16$, then the eccentricity of the ellipse $\dfrac{x^2}5+\dfrac{y^2}{b^2}=1$ is :

Let AB be a chord of length 12 of the circle $(x-2)^2+(y+1)^2=\dfrac{169}4$. If tangents drawn to the circle at points A and B intersect at the point P, then five times the distance of point P from chord AB is equal to _______.