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  • P is a point on the hyperbola <img alt="\mathrm{\frac{x^2}{a^2}-\frac{y^2}{b^2}=1}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%5Cfrac%7Bx%5E2%

P is a point on the hyperbola \mathrm{\frac{x^2}{a^2}-\frac{y^2}{b^2}=1} , N is the foot of the perpendicular from P on the transverse axis. The tangent to the hyperbola at P meets the transverse axis at T. If O is the centre of the hyperbola, then OT.ON is equal to

 

Option: 1

\mathrm{e^2}


Option: 2

\mathrm{a^2}


Option: 3

\mathrm{b^2}


Option: 4

\mathrm{b^2 / a^2}


Answers (1)

best_answer

Let \mathrm{P\left(x_1, y_1\right)} be a point on the hyperbola. Then the coordinates of N are (x, 0).

The equation of the tangent at \mathrm{(x_1, y_1)} is \mathrm{\frac{x x_1}{a^2}-\frac{y y_1}{b^2}=1}

\mathrm{\text { This meets } x \text {-axis at } T\left(\frac{a^2}{x_1}, 0\right)}

\mathrm{\therefore \quad O T \cdot O N=\frac{a^2}{x_1} \times x_1=a^2}

Posted by

manish painkra

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