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  4. If 4(x-√2)2+λ(y-√3)2=45 and (x-√2)2-4(y-√3)2=5 cut orthogonally, then λ is equal to .

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Q. If $4(x-\sqrt{2})^{2}+\lambda(y-\sqrt{3})^{2}=45$ and $(x-\sqrt{2})^{2}-4(y-\sqrt{3})^{2}=5$ cut orthogonally, then $\lambda$ is equal to ______.

Conic Sections

Solution:

Here, ellipse $\frac{(x-\sqrt{2})^{2}}{45 / 4}+\frac{(y-\sqrt{3})^{2}}{45 / \lambda}=1$
and hyperbola $\frac{(x-\sqrt{2})^{2}}{5}-\frac{(y-\sqrt{3})^{2}}{5 / 4}=1$ cut orthogonally.
So, conics will be confocal.
$\therefore \frac{45}{4}-\frac{45}{\lambda}=5+\frac{5}{4} $
$\therefore \lambda=9$