First slide

Hyperbola in conic sections

Question

Let P be a point on the hyperbola $\frac{x^{2}}{81} - \frac{y^{2}}{49} = 1$ .The tangent at P meets the transverse axis at T, N is the foot of the perpendicular from P to the transverse axis. If O is the origin, then ON.OT is equal to.

Moderate

A

81
B

49
C

-81
D

-49

Solution

Let the coordinate of P be $(9 \sec θ, 7 \tan θ)$ then $ON = 9 \sec θ$
Equation of the tangent at P is $\frac{x}{9} \sec θ - \frac{y}{7} \tan θ = 1$ which meets the transverse axis at $(9 \cos θ, 0) \Rightarrow OT = 9 \cos θ$
Then $ON . OT = 9 \sec θ \times 9 \cos θ = 81$ .

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