Circle

Let a circle $C$ touch the lines $L_{1}:4x-3y+K_{1}=0$and $L_{2}:4x-3y+K_{2}=0,K_{1},K_{2}\in\,R$. If a line passing through the centre of the circle $C$ intersects $L_{1}$at $\left(-1,2\right)$and $L_{2}$at…

Let$C$ be a circle passing through the points $A\left(2,-1\right)$and $B\left(3,4\right)$. The line segment $AB$is not a diameter of $C$. If $r$is the radius of $C$ and its centre lies on the circle…

Two tangents are drawn from a point $P$to the circle $x^{2}+y^{2}-2x-4y+4=0$, such that the angle between these tangents is $\tan\,^{-1}\left(\frac{12}{5}\right)$, where…

If the locus of the mid-point of the line segment from the point $\left(3,2\right)$ to a point on the circle, $x^{2}+y^{2}=1$is a circle of radius $r$,then $r$ is equal to

The set of all values of $a^{2}$for which the line $x+y=0$bisects two distinct chords drawn from a point $P\left(\frac{1+a}{2},\frac{1-a}{2}\right)$ on the circle $2x^{2}+2y^{2}-(1+a)x-(1-a)y=0$, is equal to :

Let $r_{1}$ and $r_{2}$ be the radii of the largest and smallest circles, respectively, which pass through the point $(-4,1)$ and having their centres on the circumference of the circle $x^{2}+y^{2}+2x+4y-4=0$. If…

If the angle of intersection at a point where the two circles with radii $5cm$ and $12cm$ intersect is $90^{\circ}$, then the length (in cm) of their common chord is:

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

Consider a circle $C_{1}:x^{2}+y^{2}-4x-2y=\alpha\,-5.$ Let its mirror image in the line $y=2x+1$ be another circle $C_{2}:5x^{2}+5y^{2}-10fx-10gy+36=0.$ Let$r$ be the radius of $C_{2}$ . Then $\alpha\,+r$ is equal to…

Let the equation $x^{2}+y^{2}+px+(1-p)y+5=0$represent circles of varying radius $r\in\,(0,5].$Then the number of elements in the set $S=${$q:q=p^{2}$and $q$ is an integer} is ___________

Let the tangents at the points $A(4,-11)$ and $B(8,-5)$ on the circle $x^{2}+y^{2}-3x+10y-15=0$, intersect at the point $C$. Then the radius of the circle, whose centre is $C$and the line joining $A$ and $B$ is its…

Let $A$be the point $\left(1,2\right)$and $B$ be any point on the curve $x^{2}+y^{2}=16$. If the centre of the locus of the point $P$, which divides the line segment $AB$in the ratio $3:2$is the point…