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  • The locus of the point of intersection of tangents to the hyperbola <img alt="\mathrm{4 x^2-9 y^2=36}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B4%20x%5E2-9%20y%5E2%3D36

The locus of the point of intersection of tangents to the hyperbola \mathrm{4 x^2-9 y^2=36} which meet at a constant angle \mathrm{\frac{\pi }{4}}, is

Option: 1

\mathrm{\left(x^2+y^2-5\right)^2=4\left(9 y^2-4 x^2+36\right)}


Option: 2

\mathrm{\left(x^2+y^2-5\right)=4\left(9 y^2-4 x^2+36\right)}


Option: 3

\mathrm{4\left(x^2+y^2-5\right)^2=\left(9 y^2-4 x^2+36\right)}


Option: 4

None of these


Answers (1)

Let the point of intersection of tangents be \mathrm{P(x_{1},y_{1}).} Then the equation of pair of tangents from \mathrm{P(x_{1},y_{1})}  to the given hyperbola is
\mathrm{\left(4 x^2-9 y^2-36\right)\left(4 x_1^2-9 y_1^2-36\right)=\left[4 x_1 x-9 y_1 y-36\right]^2\ \ \ .........(i)}
\mathrm{\text { From } S S_1=T^2 \quad \text { or } x^2\left(y_1^2+4\right)+2 x_1 y_1 x y+y^2\left(x_1^2-9\right)+\ldots . .=0\ \ \ .........(ii)}
Since angle between the tangents is \mathrm{\frac{\pi }{4}.}
\mathrm{\therefore \quad \tan (\pi / 4)=\frac{2 \sqrt{\left[x_1^2 y_1^2-\left(y_1^2+4\right)\left(x_1^2-9\right)\right]}}{y_1^2+4+x_1^2-9}.\ Hence\ locus\ of\ P\left(x_1, y_1\right)\ is}\\ \mathrm{ \left(x^2+y^2-5\right)^2=4\left(9 y^2-4 x^2+36\right)}

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Kshitij

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