Equation to locus of middle point of portion of tangent to ellipse frac{{{x^2}}}{{16}} + frac{{{y^2}

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10-2. Parabola, Ellipse, Hyperbola

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The equation to the locus of the middle point of the portion of the tangent to the ellipse $\frac{{{x^2}}}{{16}}$$+$ $\frac{{{y^2}}}{9}$ $= 1$ included between the co-ordinate axes is the curve :

$9x^2 + 16y^2 = 4 x^2y^2$

$16x^2 + 9y^2 = 4 x^2y^2$

$3x^2 + 4y^2 = 4 x^2y^2$

$9x^2 + 16y^2 = x^2y^2$

Let any tangent of ellipse is $\frac{x \cos \theta}{4}+\frac{y \sin \theta}{3}=1$

Let it meets axes at $A\left(\frac{4}{\cos \theta}, 0\right) \& B\left(0, \frac{3}{\sin \theta}\right)$

Let midpoint of $A B$ is $(h, k)$ then

$2 h=\frac{4}{\cos \theta}, 2 k=\frac{3}{\sin \theta}$

$\because \cos ^{2} \theta+\sin ^{2} \theta=1$

$\therefore \frac{16}{4 h^{2}}+\frac{9}{4 k^{2}}=1$

$\Rightarrow 16 k^{2}+9 h^{2}=4 h^{2} k^{2}$

Hence, locus is $16 y^{2}+9 x^{2}=4 x^{2} y^{2}$

Standard 11

Mathematics

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