Interaction between Circle and a Line

Find the point on the straight line, y = 2x + 11 which is nearest to the circle, $16\left({\text{x}}^2+{\text{y}}^2\right)+32\text{x}-8\text{y}-50=0$

One of the diameters of the circle circumscribing the rectangle $ABCD$ is $4y=x+7$. If $A$ and $B$ are the points $\left(- 3,4\right)$ and $\left(5, 4\right)$, respectively, then the area of the rectangle is 

The points of intersection of the line  $\mathrm{}\mathrm{}4x-3y-10=0$ and the circle  $x^2+y^2-2x+4y-20=0$ are:

Let $O$ be the center of the circle $x^2+y^2=r^2,$ where $r>\frac{\sqrt{5}}{2}.$ Suppose $PQ$ is a chord of this circle and the equation of the line passing through $P$ and $Q$ is  $2x+4y=5$. If the center of the circumcircle of the triangle $OPQ$ lies on the line $x+2y=4,$ then the value of $r$ is ______

Equation of the circle, which is the mirror image of the circle $x^2+y^2-2x=0$ with respect to the line $x+y=2$, is

If one of the diameters of the circle $x^2+y^2-2x-6y+6=0$ is a chord to the circle with centre $\left(2, 1\right)$, then the radius of the circle is

The circle $x^2+y^2-3x-4y+2=0$ cuts $x$-axis at

The locus of centre of a circle which passes through the origin and cuts off a length of $4$ unit from the line $x=3$ is

Radius of circle in which a chord length $\sqrt{2}$ makes an angle $\frac{\pi }{2}$ at the centre, is

Let $A, B, C$ be three points on a circle of radius $1$ such that $\angle ACB=\frac{\pi }{4}$. Then the length of the side $AB$ is

Find the length of the chord intercepted by the circle $x^2+y^2-x+3y-22=0$ on the line $y=x-3$.

Let $S_1\equiv {\left(x+3\right)}^2+y^2=9$ has centre $C_1$ and if the circle $x^2+y^2+2x-3=0$ cuts the circle $S_1$ at $B$ and $C$, then line segment $BC$ subtends an angle at the centre $C_1$ will be

Consider
$\begin{array}{l}{L}_1:2x+3y+p-3=0\\ L_2:2x+3y+p+3=0\end{array}$
where, $p$ is a real number and
$C:x^2+y^2-6x+10y+30=0$

Statement I: If line $L_1$ is a chord of circle $C$, then line $L_2$ is not always a diameter of circle $C$.

Statement II: If line $L_1$ is a diameter of circle $C$, then line $L_2$ is not a chord of circle $C$.

The length of the intercept made by the circle $x^2+{\mathrm{y}}^2=2$ on the line $y=2x+1$ is

Locus of the center of the circles with $(2, 3)$ as the midpoint of the chord $5x+2y=16$ is:

Find points of intersection of the line $4x-3y-10=0$  and the circle $x^2+y^2-2x+4y-20=0$

Let $S$ be the region consisting of points $(x, y)$ satisfying the inequality $x^2+y^2-6x-8y+21\le 0$
and $x,y\in R.$ Let minimum and maximum value of $\frac{y}{x}$ be $m \& M$ respectively, where $(x,y)\in S.$
Then value of $14\mathrm{Mm}$ is.

One of the diameters of circle circumscribing the rectangle $\mathrm{ABCD}$ is $4y=x+7$. If $\mathrm{A}$ and $\mathrm{B}$ are the points $(-3,4)$ and $(5,4)$ respectively then the area of the rectangle is (in square units)

The sum of the slopes of all possible chords of the circle $x^2+y^2=100,$ which passes through $\left(7,1\right)$ and also divide the circumference of this circle into two arcs whose lengths are in the ratio $2:\mathrm{1,}$ is equal to