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10-2. Parabola, Ellipse, Hyperbola
hard
If any tangent to the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ cuts off intercepts of length $h$ and $k$ on the axes, then $\frac{{{a^2}}}{{{h^2}}} + \frac{{{b^2}}}{{{k^2}}} = $
A
$0$
B
$1$
C
$-1$
D
None of these
Solution
(b) The tangent at $(a\cos \theta ,\,b\sin \theta )$ to the ellipse is
$\frac{{(a\cos \theta )x}}{{{a^2}}} + \frac{{(b\sin \theta )y}}{{{b^2}}} = 1$ or $\frac{x}{{(a/\cos \theta )}} + \frac{y}{{(b/\sin \theta )}} = 1$
$\therefore $ Intercepts are, $h = \frac{a}{{\cos \theta }},\,\,k = \frac{b}{{\sin \theta }}$
==> $\frac{{{a^2}}}{{{h^2}}} + \frac{{{b^2}}}{{{k^2}}} = 1$.
Standard 11
Mathematics
