Question

The chord of contact of tangents drawn from a point on the circle $x^2 + y^2 = 16$ to the circle $x^2 + y^2 = r^2$ touches the circle $x^2 + y^2 = 1$, then the value of $\lvert r\rvert$ is equal to

Answer 1

2

Answer 2

Solution Approach

1. Let $P(x_1,y_1)$ be a point on the circle $x^2+y^2=16$.
   $\therefore x_1^2 + y_1^2 = 16.$

2. The equation of the chord of contact of tangents drawn from $P$ to the circle $x^2+y^2=r^2$ is

   $$xx_1 + yy_1 = r^2.$$

3. This line touches the circle $x^2+y^2=1$.

   • Distance from center $(0,0)$ to the line $xx_1+yy_1=r^2$ must equal the radius $1$.
   • Distance formula: $\displaystyle \frac{\lvert r^2 \rvert}{\sqrt{x_1^2 + y_1^2}} = 1$.

4. Substitute $x_1^2 + y_1^2 = 16$:

   $$\frac{\lvert r^2\rvert}{\sqrt{16}} = 1 \quad\Longrightarrow\quad \frac{\lvert r^2\rvert}{4} = 1 \quad\Longrightarrow\quad r^2 = 4.$$

5. Hence,

   $$\lvert r\rvert = 2.$$