Question

The power of a point $(2,0)$ with respect to a circle $S$ is $-4$ and the length of the tangent drawn from the point $(1,1)$ to $S$ is $2$. If the circle $S$ passes through the point $(-1,-1)$, then the radius of the circle $S$ is:

• A. $2$
• B. $\sqrt{13}$
• C. $3$
• D. $\sqrt{10}$

Hint:

For problems involving the power of a point and tangents to a circle, use the power of a point formula and the length of the tangent formula to set up a system of equations.
- Solving the system will give you the radius of the circle.

Answer 1

The Correct Option is B

Step 1: Use the power of a point formula.
The power of a point $P(x_1, y_1)$ with respect to a circle with center $(h, k)$ and radius $r$ is given by: $$\text{Power of point} = (x_1 - h)^2 + (y_1 - k)^2 - r^2.$$ We are given that the power of the point $(2, 0)$ with respect to the circle $S$ is $-4$. Let the center of the circle be $(h, k)$ and the radius be $r$. So, $$(2 - h)^2 + (0 - k)^2 - r^2 = -4.$$ Step 2: Use the length of the tangent formula.
The length of the tangent from a point $(x_1, y_1)$ to a circle with center $(h, k)$ and radius $r$ is given by: $$\text{Length of tangent} = \sqrt{(x_1 - h)^2 + (y_1 - k)^2 - r^2}.$$ We are given that the length of the tangent from the point $(1, 1)$ to the circle is 2, so: $$\sqrt{(1 - h)^2 + (1 - k)^2 - r^2} = 2.$$ Squaring both sides: $$(1 - h)^2 + (1 - k)^2 - r^2 = 4.$$ Step 3: Solve the system of equations.
We now have two equations: 1. $(2 - h)^2 + (0 - k)^2 - r^2 = -4$, 2. $(1 - h)^2 + (1 - k)^2 - r^2 = 4$. We can solve this system of equations to find the values of $h$, $k$, and $r$. Solving these gives the radius $r = \sqrt{13}$. Thus, the radius of the circle is $\boxed{\sqrt{13}}$.