Question

The chord of contact of the tangents drawn from a point on the circle, $x^2 + y^2 = a^2$ to the circle $x^2 + y^2 = b^2$ touches the circle $x^2 + y^2 = c^2$, where a, b, c > 0 then a, b, c are in :

• A. Arithmetic Progression
• B. Geometric Progression
• C. Harmonic Progression
• D. None of these

Answer 1

Answer: (B)

Geometric Progression

Answer 2

Let $P(x_1, y_1)$ be a point on the circle $x^2 + y^2 = a^2$. This implies that $x_1^2 + y_1^2 = a^2$.

The equation of the chord of contact of tangents drawn from $P(x_1, y_1)$ to the circle $x^2 + y^2 = b^2$ is given by $xx_1 + yy_1 = b^2$.

This chord of contact touches the circle $x^2 + y^2 = c^2$. The condition for a line $Ax + By + C = 0$ to touch a circle $x^2 + y^2 = r^2$ is that the perpendicular distance from the center $(0,0)$ to the line equals the radius $r$.

For the line $xx_1 + yy_1 - b^2 = 0$ and the circle $x^2 + y^2 = c^2$ (with radius $c$), the perpendicular distance is: $d = \frac{|x_1(0) + y_1(0) - b^2|}{\sqrt{x_1^2 + y_1^2}} = \frac{|-b^2|}{\sqrt{x_1^2 + y_1^2}}$ Since $b > 0$, $|-b^2| = b^2$. $d = \frac{b^2}{\sqrt{x_1^2 + y_1^2}}$

For tangency, $d = c$: $\frac{b^2}{\sqrt{x_1^2 + y_1^2}} = c$

Since $P(x_1, y_1)$ is on $x^2 + y^2 = a^2$, we have $\sqrt{x_1^2 + y_1^2} = \sqrt{a^2} = a$ (as $a > 0$). Substituting this into the equation: $\frac{b^2}{a} = c$ $b^2 = ac$

This is the condition for $a, b, c$ to be in Geometric Progression.