Question
The locus of the point of intersection of the tangent to the circle $x^{2} + y^{2} = a^{2}$, which include an angle of $45^{\circ}$ is the curve $\left( x^{2} + y^{2} \right)^{2} = \lambda a^{2}\left( x^{2} + y^{2} - a^{2} \right)$. The value of $\lambda$ is:
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Solution
Sol. [Key Concept: Tangents to $x^{2} + y^{2} = a^{2}$ from a point at distance $L$: $\tan\theta = \frac{R}{L}$ with $2\theta{= 45}^{\circ}$ gives the locus.]
$$tan2\theta = \frac{2tan\theta}{1 - \tan^{2}\theta}$$
$$\begin{matrix} & 1 = \frac{2(R/L)}{1 - \left( R^{2}/L^{2} \right)} = \frac{2RL}{L^{2} - R^{2}} \\ & \ = \frac{2a\sqrt{x_{1}^{2} + y_{1}^{2} - a^{2}}}{x_{1}^{2} + y_{1}^{2} - a^{2} - a^{2}} \\ & \left( x^{2} + y^{2} - 2a^{2} \right)^{2} = 4a^{2}\left( x^{2} + y^{2} - a^{2} \right) \\ & \left( x^{2} + y^{2} \right)^{2} + 4a^{4} - 4a^{2}\left( x^{2} + y^{2} \right) \\ & \ = 4a^{2}\left( x^{2} + y^{2} - a^{2} \right) \\ & \left( x^{2} + y^{2} \right)^{2} + 8a^{4} = 8a^{2}\left( x^{2} + y^{2} \right) \\ & \ \left( x^{2} + y^{2} \right)^{2} = 8a^{2}\left( x^{2} + y^{2} - a^{2} \right) \\ & \ \Rightarrow \ \lambda = 8 \end{matrix}$$
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