Tangents to the ellipse $b^{2} x^{2} + a^{2} y^{2} = a^{2} b^{2}$ make complementary angles with the major axis. Then the locus of their point of intersection is

x^{2} + y^{2} = a^{2} - b^{2}

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x^{2} - y^{2} = a^{2} + b^{2}

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x^{2} - y^{2} = a^{2} - b^{2}

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x^{2} + y^{2} = a^{2} + b^{2}

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Solution

The correct option is B $x^{2} - y^{2} = a^{2} - b^{2}$
Let the point of intersection of tangent be $(h, k)$

Then equation of tangent with slope $m$ is given by $y = m x + \sqrt{a^{2} m^{2} + b^{2}}$

$(h, k)$ lie on tangent

then $k = m h + \sqrt{a^{2} m^{2} + b^{2}}$

$(a^{2} - h^{2}) m^{2} + 2 h k m + b^{2} - k^{2} = 0$

since, tangents are complementary

therefore, $m_{1} m_{2} = 1$

$\Rightarrow \frac{b^{2} - k^{2}}{a^{2} - h^{2}} = 1$

Therefore, $h^{2} - k^{2} = a^{2} - b^{2}$
$∴ x^{2} - y^{2} = a^{2} - b^{2}$

Ans: C

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