The number of points on rectangular hyperbola $xy = c^{2}$ from which two tangents drawn to ellipse

Question

The number of points on rectangular hyperbola $xy = c^{2}$ from which two tangents drawn to ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ where $0 < b < a < c$ are ⟂ ar to each other is :

Accepted Answer

Sol. (a) Point of hyperbola $\left( ct,\frac{c}{t} \right)$ lie on director circle $x^{2} + y^{2} = a^{2} + b^{2}$ of ellipse$\Rightarrow a^{2} + b^{2} = c^{2}\left( t^{2} + \frac{1}{t^{2}} \right) \geq 2c^{2}$ which is impossible$$\because\ a^{2} + b^{2} < c^{2} + c^{2} = 2c^{2}$$$$(\because b < a < c)$$

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