Number of Tangents from a Point on a Circle

Tangents are drawn for the circumcircle of an equilateral $△ABC$ with a side length of $4 \mathrm{cm}$ at the vertices $A, B,$ and $C$ of the triangle. The three tangents intersect at $P, Q,$ and $R$ to form a $△PQR$. Find the area of $△PQR$.

People of village want to construct a road nearest to a circular village Rampur. The road cannot pass through the village . But the people want the road should be at the shortest distance from the cente of the village.

Suppose , the road start from point $O$, which is outside the circular village and touch the boundary of the circular village at point A such that $OA=20 cm$ and also , the straight distance of the point $O$ from the centre $C$ of the village is $25 cm.$

Find the shortest distance of the road from the centre of the village.

If a point is inside the circle, how many tangents can be drawn from that point? Also, if two circles are externally and they do not touch, then find the number of common tangents.

In the diagram, point $B$ is a point of tangency. Find the radius of the circle.

What is the angle between the tangents if the corresponding radii are inclined at an angle of $90°?$

The length of tangent segment from $(2,6)$ to the circle $x^2+y^2=25$ is :

Find the length of tangent segment of the circle $4x^2+4y^2+16x-24y+3=0$ from the point $(-1,-1).$

Find the length of the tangent of circle from an external point $Q$ of distance $25 cm.$ from centre of circle $O$of radius $7 cm?$

If from a point P only one tangent can be drawn to a circle, then P is on the circle.

Number of tangents can be drawn from the point outside the circle.

Find the position of the points $(2,3)$ and $(3,1)$ with respect to the circle ${\left(x+1\right)}^2+{\left(y-1\right)}^2=18$.

Find the position of the points $(1,4)$ and $(4,1)$ with respect to the circle ${\left(x-2\right)}^2+{\left(y+1\right)}^2=20$.

Find the position of the points $(1,2)$ and $(6,0)$ with respect to the circle ${\left(x-2\right)}^2+{\left(y+1\right)}^2=16$.

Given the three lines are tangent, find the value of $x+y$ in the figure.

What is the difference of perimeter of $∆ABC$ and $∆ADE$?

Prove that the point $(1, - 1)$ lies within the circle $x^2 + y^2 - 4x + 6y + 4 = 0.$

Find the position of the point $\left(2,3\right)$  with respect to the circle $x^2+y^2-6x-4y=0$.

Find the position of the point $\left(3,4\right)$  with respect to the circle $x^2+y^2=9$

Find the position of the point $\left(5,12\right)$  with respect to the circle $x^2+y^2=169.$

The number of tangents to a circle drawn from a point outside the circle is:

How many tangents can be constructed from a point inside the circle?