Circles

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

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

Let a circle $C$ have its centre in the first quadrant, intersect the coordinate axes at exactly three points and cut off equal intercepts from the coordinate axes. If the length of the chord of $C$ on the line $x+y=1$ is $\sqrt{14}$, then the square of the radius of $C$ is ________.

Let the circles $C_1:|z|=r$ and $C_2:|z-3-4i|=5$, $z\in\mathbb C$, be such that $C_2$ lies within $C_1$. If $z_1$ moves on $C_1$, $z_2$ moves on $C_2$ and $\min|z_1-z_2|=2$, then $\max|z_1-z_2|$ is equal to :

Let a circle pass through the origin and its centre be the point of intersection of two mutually perpendicular lines $x+(k-1)y+3=0$ and $2x+k^2y-4=0$. If the line $x-y+2=0$ intersects the circle at the points A and B, then $(AB)^2$ is equal to :

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 ________.

Suppose that two chords, drawn from the point $(1,2)$ on the circle $x^2+y^2+x-3y=0$ are bisected by the $y$-axis. If the other ends of these chords are R and S, and the mid point of the line segment RS is $(\alpha,\beta)$, then $6(\alpha+\beta)$ is equal to:

Let $A=\begin{bmatrix}1&2&7\\4&-2&8\\3&8&-7\end{bmatrix}$ and $\det(A-\alpha I)=0$, where $\alpha$ is a real number. If the largest possible value of $\alpha$ is $p$, then the circle $(x-p)^2+(y-2p)^2=320$, intersects the co-ordinate axes at

Let $P(3\cos\alpha,2\sin\alpha)$, $\alpha\ne0$, be a point on the ellipse $\dfrac{x^2}9+\dfrac{y^2}4=1$, Q be a point on the circle $x^2+y^2-14x-14y+82=0$ and R be a point on the line $x+y=5$ such that the centroid of the triangle PQR is $\left(2+\cos\alpha,\ 3+\dfrac23\sin\alpha\right)$. Then the sum of the ordinates of all possible points R is:

Let P be a moving point on the circle $x^2+y^2-6x-8y+21=0$. Then, the maximum distance of P from the vertex of the parabola $x^2+6x+y+13=0$ is equal to:

Let $y=y(x)$ be the solution of the differential equation $x\sin\left(\dfrac yx\right)dy=\left(y\sin\left(\dfrac yx\right)-x\right)dx$, $y(1)=\dfrac\pi2$ and let $\alpha=\cos\left(\dfrac{y\left(e^{12}\right)}{e^{12}}\right)$. Then the number of integral value of $p$, for which the equation $x^2+y^2-2px+2py+\alpha+2=0$ represents a circle of radius $r\le6$, is ________.

Let the point P be the vertex of the parabola $y=x^2-6x+12$. If a line passing through the point P intersects the circle $x^2+y^2-2x-4y+3=0$ at the points R and S, then the maximum value of $(PR+PS)^2$ is :

Let the set of all values of $k\in\mathbb R$ such that the equation $z\left(\bar z+2+i\right)+k(2+3i)=0$, $z\in\mathbb C$, has at least one solution, be the interval $[\alpha,\beta]$. Then $9(\alpha+\beta)$ is equal to:

Let the centre of the circle $x^2+y^2+2gx+2fy+25=0$ be in the first quadrant and lie on the line $2x-y=4$. Let the area of an equilateral triangle inscribed in the circle be $27\sqrt3$. Then the square of the length of the chord of the circle on the line $x=1$ is ________.

Let C be a circle having centre in the first quadrant and touching the $x$-axis at a distance of 3 units from the origin. If the circle C has an intercept of length $6\sqrt3$ on $y$-axis, then the length of the chord of the circle C on the line $x-y=3$ is :

Let the line $x-y=4$ intersect the circle $C:(x-4)^2+(y+3)^2=9$ at the points Q and R. If $P(\alpha,\beta)$ is a point on C such that $PQ=PR$, then $(6\alpha+8\beta)^2$ is equal to ________.

Consider the circle $C:x^2+y^2-6x-8y-11=0$. Let a variable chord AB of the circle C subtend a right angle at the origin. If the locus of the foot of the perpendicular drawn from the origin on the chord AB is the circle $x^2+y^2-\alpha x-\beta y-\gamma=0$, then $\alpha+\beta+2\gamma$ is equal to ________.

Let $\mathbb{R}$ denote the set of all real numbers. Let $z_1 = 1 + 2i$ and $z_2 = 3i$ be two complex numbers, where $i = \sqrt{-1}$. Let

$$S = \{(x, y) \in \mathbb{R}\times\mathbb{R} : |x + iy - z_1| = 2|x + iy - z_2|\}.$$

Then which of the following statements is (are) TRUE?

Let $P(x_1, y_1)$ and $Q(x_2, y_2)$ be two distinct points on the ellipse $\dfrac{x^2}{9} + \dfrac{y^2}{4} = 1$ such that $y_1 > 0$, and $y_2 > 0$. Let $C$ denote the circle $x^2 + y^2 = 9$, and $M$ be the point $(3, 0)$.

Suppose the line $x = x_1$ intersects $C$ at $R$, and the line $x = x_2$ intersects $C$ at $S$, such that the $y$-coordinates of $R$ and $S$ are positive. Let $\angle ROM = \dfrac{\pi}{6}$ and $\angle SOM = \dfrac{\pi}{3}$, where $O$ denotes the origin $(0, 0)$. Let $|XY|$ denote the length of the line segment $XY$.

Then which of the following statements is (are) TRUE?

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