Circle Geometry: Chord of Contact Analysis

We need to find the value of $m + n + p + 10$ given the relationship derived from the chord of contact touching circles.

1. Identify the Point and Chord of Contact

• Let the point from which tangents are drawn be $P(x_0, y_0)$. Since $P$ lies on the circle $x^2 + y^2 = a^2$, we have:

  $x_0^2 + y_0^2 = a^2$

• The equation of the chord of contact of tangents drawn from $P(x_0, y_0)$ to the circle $x^2 + y^2 = b^2$ is given by the standard form:

  $xx_0 + yy_0 = b^2$

2. Apply Tangency Condition

• The chord of contact line, $xx_0 + yy_0 - b^2 = 0$, touches the circle $x^2 + y^2 = c^2$.
• For a line $Ax + By + C = 0$ to touch a circle $x^2 + y^2 = r^2$, the distance from the center $(0, 0)$ to the line must equal the radius $r$.
• Distance formula: $d = \frac{|A(0) + B(0) + C|}{\sqrt{A^2 + B^2}} = r$
• In our case, $A = x_0$, $B = y_0$, $C = -b^2$, and $r = c$.
• Applying the condition:

  $\frac{|0 \cdot x_0 + 0 \cdot y_0 - b^2|}{\sqrt{x_0^2 + y_0^2}} = c$

  $\frac{|-b^2|}{\sqrt{x_0^2 + y_0^2}} = c$

  $\frac{b^2}{\sqrt{x_0^2 + y_0^2}} = c$

3. Derive Radii Relationship

• Substitute $x_0^2 + y_0^2 = a^2$ into the equation:

  $\frac{b^2}{\sqrt{a^2}} = c$

  $\frac{b^2}{a} = c$

• Rearranging gives the relationship between the radii:

  $b^2 = ac$

4. Determine Exponents m, n, p

• We are given the condition $b^m = a^n c^p$, where $m, n, p \in \mathbb{N}$.
• Our derived relationship is $b^2 = ac$, which can be written as $b^2 = a^1 c^1$.
• Comparing $b^m = a^n c^p$ with $b^2 = a^1 c^1$, we find the exponents:

  $m = 2$

  $n = 1$

  $p = 1$

• These values ($2, 1, 1$) are natural numbers, satisfying the condition $m,n,p \in \mathbb{N}$.

5. Calculate the Final Value

• The question asks for the value of $m + n + p + 10$.
• Substitute the values:

  $m + n + p + 10 = 2 + 1 + 1 + 10 = 14$