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Question

If the tangents are drawn from (3, 2) to the hyperbola x29y2=9. Find the area of the triangle (in sq. unit) that these tangents form with their chord of contact.

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Solution

Equation of the chord of contact of the tangents drawn from the point (3, 2) to the hyperbola is

T=0

3x9×2y=9

x-6y=3 .........(1)

Equation of the hyperpola is x29y2=9 ..........(2)

Solving equation of chord of contact of hyperpola with the equation of hyperpola.

substituting x=3+6y in the equation of hyperpola

(3+6y)29y2=9

(1+2y)2y2=1

1+4y2+4yy2=1

3y2+4y=0

y(3y+4)=0

y=0 & y=43

When y=0,x=3

when y=43,x=6(43)+3

x=8+3=5

point of intersection being (3,0) & (5,43)

Area of triangle formed by points (3,2),(3,0) & (5,43) is

=12∣ ∣ ∣1113352043∣ ∣ ∣

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