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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.


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∣ ∣ ∣

Mathematics

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