The chord of contact of tangents drawn from a point on the circle x² + y² = a² to the circle x² +
y² = b² touches the circle x² + y² = c² such that$$b^m=a^n c^p,$$​ where m, n, p, ∈ N. Find the value of
m + n + p + 10.

Correct option is C

Given:

​The chord of contact of tangents drawn from a point on the circle x² + y² = a²

to the circle x² +​y² = b² touches the circle x² + y² = c²

such that $$b^m=a^n c^p$$, where m, n, p, ∈ N. 

​m + n + p + 10. 

Solution:

Let there is a point P(h, k) on the circle$$x^2 + y^2 = a^2.$$​​

(h, k) will satisfy the equation$$x^2 + y^2 = a^2$$​.

=>$$h^2 + k^2 = a^2$$

​​

The equation of the chord of the contact of tangents drawn from the point P(h, k) to the circle$$x^2 + y^2 = b^2$$​ will be

​$$hx + ky = b^2$$​​

The perpendicular distance of this tangent from the center of the circle $$x^2 + y^2 = c^2$$ will be equal to the radius of the circle.

​$$\left| \frac{-b^2}{\sqrt{h^2 + k^2}} \right| = c$$ 

$$\left| \frac{-b^2}{a} \right| = c$$ 

$$b^2 = ac$$​​

Comparing it with $$b^m=a^n c^p$$​​

=> p = 1, m =2 ,n = 1 

so, ​m + n + p + 10.  =  10+1+1=+2   

                                         =14