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  • If  and <img alt="\mathrm{C_3}" src="https://entrancecorner.oncodecog

If \mathrm{C_1, C_2} and \mathrm{C_3} belong to a family of circles through the points  \mathrm{\left(x_1, y_1\right)} and \mathrm{\left(x_2, y_2\right)} , then the ratio of the lengths of the tangents from any point \mathrm{C_{1}} to the circles \mathrm{C_{2}} and \mathrm{C_{3}} is 

Option: 1

vary according to point \mathrm{\left(x_1, y_1\right) \text { and }\left(x_2, y_2\right)}


Option: 2

constant


Option: 3

vary according to point on 


Option: 4

None of these


Answers (1)

best_answer

Equations of circles \mathrm{C_1, C_2} and \mathrm{C_3} through \mathrm{A\left(x_1, y_1\right)} and \mathrm{B\left(x_2, y_2\right)} are given by 

\mathrm{\begin{aligned} & \underbrace{\left(x-x_1\right)\left(x-x_2\right)+\left(y-y_1\right)\left(y-y_2\right)}_{f(x, y)} \\ & +\lambda_r \underbrace{\left(y-y_1-\frac{y_2-y_1}{x_2-x_1}\left(x-x_1\right)\right)}_{g(x, y)}=0 ; r=1,2,3 \\ & f(x, y)+\lambda, g(x, y)=0 ; r=1,2,3 \\ & \end{aligned}}

Consider point P(h,k) on the circle \mathrm{C_1} 

\mathrm{\therefore \quad f(h, k)+\lambda_1 g(h, k)=0}

Let \mathrm{T_2} and \mathrm{T_3} be the lengths of the tangents from P to \mathrm{C_2} and \mathrm{C_3} respectively.

\mathrm{\begin{aligned} \therefore \frac{T_2}{T_3} & =\frac{\sqrt{f(h, k)+\lambda_2 g(h, k)}}{\sqrt{f(h, k)+\lambda_3 g(h, k)}} \\ & =\frac{\sqrt{-\lambda_1 g(h, k)+\lambda_2(h, k)}}{\sqrt{-\lambda_1 g(h, k)+\lambda_3 g(h, k)}}=\sqrt{\frac{\lambda_2-\lambda_1}{\lambda_3-\lambda_1}} \end{aligned}}

Clearly, this ratio is independent of the choice (h,k) and hence a constant.

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