Tangent and Normal to a Circle

How many tangents can be drawn to a circle from a point on the same circle? Justify your answer.

Find the equation of the normal to the circle $x^2+y^2=30$ in the parametric form, where the parameter is $\theta =120°$.

The tangent at $A(-1,2)$ on the circle $x^2+y^2-4x-8y+7=0$ touches the circle $x^2+y^2+4x+6y=0$ at $B$. Then, a point of trisection of $AB$ is

The equation of the normal to the circle $x^2+y^2=2x$ which is parallel to the straight line $x+2y=3$ is given by

A circle has a diameter having the coordinates as $\left(3,7\right)$ and $\left(9,1\right)$, also this circle touches the line $x+y=4$, find the coordinates of point of contact :

The line $l: 4x-y=2$  touches a circle $\omega$ at $(1,2).$ The equation of circle $\omega ,$ if it passes through $(-2,3),$ is $a{x}^2+b{y}^2+cx+dy+e=0$. Evaluate $|c-d+e-ab|$.

The tangents of two points $A$ and $B$ on the circle with centre $O$ intersect at a point $P.$ If in quadrilateral $PAOB,$ $\angle AOB:\angle APB=5:1,$ then the measure of $\angle APB$ is given by

The locus of the point of intersection of two perpendicular tangents of a circle is called

The locus of the point of intersection of tangents drawn to a circle $x^2+y^2=4$ if the difference between their parameters at the point of contact is ${120}^o$

The normal to the circle given by $x^2+y^2-6x+8y-144=0$ at $(8,8)$ meets the circle again at the point

The normal drawn at $P\left(-1, 2\right)$ on the circle $x^2+y^2-2x-2y-3=0$ meets the circle at another point $Q$ . Then the coordinates of $Q$ are

If $53x-54y+17=0$ and $106x-108y=-4$ are the tangents of the circle, then radius of the circle is = ........

How many common tangents can be drawn to the following circles $x^2+y^2=6x$ and $x^2+y^2+6x+2y+1=0$

The number of distinct tangents that can be drawn from the origin to the circle $x^2+y^2=2\left(x+y\right)$ is

The angle between the tangents drawn at the points $(5, 12)$ and $(12, -5)$ to the circle $x^2+y^2=169$ is

The coordinates of the centre of the smallest circle passing through the origin and having $y=x+1$ as a diameter are

If the line $y\cos \mathrm{\alpha }=x\sin \mathrm{\alpha }+a\cos \mathrm{\alpha }$ be a tangent to the circle $x^2+y^2=a^2$, then

The equation of the tangent to circle $5x^2+5y^2=1$, parallel to line $3x+4y=1$ are

The equation of the tangent to circle $5x^2+5y^2=1$, parallel to line $3x+4y=1$ are

The equation of the tangent to circle $5x^2+5y^2=1$, parallel to line $3x+4y=1$ are