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

If a circle of centre $\left({\alpha }^2,0\right)$ and radius $R$ touches by the line $y=x$ and cuts off a chord of length $2$ units along the line $\sqrt{3}y-x=0$. Then the value of $2{\alpha }^2{R}^2$ is equal to

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 sum of square of the length of the chords intercepted by the line $x+y=n; n\in N$ on the circle $x^2+y^2=4$.

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

Set of values of $m$ for which two points $P$ and $Q$ lie on the line $y=mx+8$ so that $\angle APB=\angle AQB=\frac{\pi }{2}$ where $A\equiv (-4,0),B\equiv (4,0)$ is -

If two distinct chords drawn from the point $\left(2\sqrt{2}\sin \theta ,\frac{1}{\sqrt{2}}\right)$ to the circle $x^2+y^2=2\sqrt{2}\sin \theta x+\frac{1}{\sqrt{2}}y,$ ($\theta$ is a parameter) are bisected by $x$ -axis, then the exhaustive set in which $\theta$ lies is

Three parallel chords of a circle have lengths $2, 3, 4$ and subtend angles $\alpha , \beta , \alpha +\beta$ at the centre respectively (given $\alpha +\beta <\pi$), then $\cos \alpha$ is equal to

Let $C_1$ and $C_2$ are two circles $x^2+y^2+10x-24y-87=0$ and $x^2+y^2-10x-24y+153=0$ respectively. Let $m$ be the smallest positive value of a for which the line $y=ax$ contains the centre of a circle that is internally tangent to $C_1$ and externally tangent to $C_2$, then the value of $\left[m\right]$ is (where $[·]$ denotes greatest integer function)

If points $(0, 0), (\mathrm{\lambda }, 0),(0, \mathrm{\lambda })$ and $\left(\mu -1, \mu +3\right)$ are always concyclic $\forall \mu \in R$ (where $\mathrm{\lambda }\in R$ and $\mathrm{\lambda }\ne 0$, then minimum value of $\left|\mathrm{\lambda }\right|$ is

If the variable line $3x-4y+k=0$ lies between the circles $x^2+y^2-2x-2y+1=0$ and $x^2+y^2-16x-2y+61=0$ without intersecting or touching either circle, then range of $k$ is $\left(a,b\right)$ where $a, b\in I$. Then the value of $\left(b-a\right)$, is