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The area of triangle formed by the tange...

The area of triangle formed by the tangents from the point (3, 2) to the hyperbola `x^2-9y^2=9` and the chord of contact w.r.t. the point (3, 2) is_____________

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The correct Answer is:
8
The hyperbola is
`x^(2)-9y^(2)=9`
`"or "(x^(2))/(9)-(y^(2))/(1)=1`
The equation of tangent is
`y=mx pm sqrt(a^(2)m^(2)-b^(2))" (1)"`
It passes through (3,2). Therefore,
`2=3mpmsqrt(9m^(2)-1)`
`"or "4+9m^(2)-12m=9m^(2)-1`
Solving, we get the values of m as
`m_(1)=(5)//(12) and m_(2)=oo`
Therefore, equations of tangents are
`5x-12y+9=0" (2)"`
and `x - 3 = 0" "(3)`
Now, the equation of chord of contact w.r.t. point P(3,2) is
T = 0
`"or "3x-18y=9`
`"or "x-6y=3" (3)"`
Solving (2) and (4), we get
`x= -5, y = -(4)/(3)`
Solving (3) and (4), we get
x = 3, y = 0
Now, the vertices of triangle are `(3, 2),(3,0), and (-5,-4//3)`.
Therefore,
`"Area"=(1)/(2)|(3,2,1),(3,0,1),(-5,-4//3,1)|`
`="8. sq. units"`
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