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Question: P is a point on the hyperbola \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1,N\) is the foot of the...

P is a point on the hyperbola x2a2−y2b2=1,N\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

A

e2e^{2}

B

a2a^{2}

C

b2b^{2}

D

b2a2\frac{b^{2}}{a^{2}}

Answer

a2a^{2}

Explanation

Solution

Let P(x1,y1)P(x_{1},y_{1}) be a point on the hyperbola. Then the co-ordinates of N are (x1,0)(x_{1},0).

The equation of the tangent at (x1,y1)(x_{1},y_{1}) is xx1a2−yy1b2=1\frac{xx_{1}}{a^{2}} - \frac{yy_{1}}{b^{2}} = 1

This meets x-axis at T(a2x1,0)T\left( \frac{a^{2}}{x_{1}},0 \right);

∴\therefore OT.ON=a2x1×x1=a2OT.ON = \frac{a^{2}}{x_{1}} \times x_{1} = a^{2}