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10-2. Parabola, Ellipse, Hyperbola
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A tangent to the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ intersect the co-ordinate axes at $A$ and $B,$ then locus of circumcentre of triangle $AOB$ (where $O$ is origin) is
A
$\frac{16}{x^2}+\frac{25}{y^2}=1$
B
$16x^2 + 25y^2 = 4$
C
$\frac{25}{x^2}+\frac{16}{y^2}=4$
D
$\frac{25}{x^2}+\frac{16}{y^2}=1$
Solution
Any tangent to the ellipse is
$\frac{x \cos \theta}{5}+\frac{y \sin \theta}{4}=1$
$\therefore $ $A = (5\sec \theta ,0),B = (0,4\cos ec\theta )$
Let $(\mathrm{h}, \mathrm{k})$ be the circumcentre of $\Delta \mathrm{A} \mathrm{OB},$ then
$2 \mathrm{h}=\frac{5}{\cos \theta}$ and $2 \mathrm{k}=\frac{4}{\sin \theta}$
$\therefore $ Locus of $(\mathrm{h}, \mathrm{k})$ is $\frac{25}{\mathrm{x}^{2}}+\frac{16}{\mathrm{y}^{2}}=4.$
Standard 11
Mathematics
