Find the equations of the tangents to the hyperbola `x^2=9y^2=9` that are drawn from (3, 2).

Options

A. equation of one of the tangents is x = 3
B. equation of one of the tangents is `5x-12y+9=0`
C. the area of triangle that these tangents form with their chord of contact is 12 sq. untis
D. the area of triangle that these tangents form with their chord of contact is 8 sq. units

Answer

A, B, D

Explanation

Given hyperbola is
`x^(2)-9y^(2)=9`
`"or "(X^(2))/(9)-(y^(2))/(1)=1`
Equation of tangent is
`y=mx pm sqrt(a^(2)m^(2)-b^(2))`
This tangent passes through `(3, 2)`.
So, `2=3m pm sqrt(9m^(2)-1)`
`"rArr "4+9m^(2)-12m=9m^(2)-1`
`therefore" "m_(1)=5//12 and m_(2)=oo`
So, equations of tangents are
`y-2=(5//12)(x-3)" (1)"`
`rArr" "5x-12y+9=0" (2)"`
and x = 3

Now, equation of chord of contact w.r.t. point P(3, 2) is
T = 0
`rArr" "3x-18y=9`
`rArr" "x-6y=3" (3)"`
Solving (1) and (3), we get `x=-5, y=-(4)/(3)`.
Solving (2) and (3), we get x = 3, y = 0.
Thus, vertices of triangle are `(3, 2), (3, 0), (-5, -4/3)`.
`"Area of triangle"=(1)/(2)|(3,2,1),(3,0,1),(-5,-(4)/(3),1)|`
`=(1)/(2)xx|3((4)/(3))-2(3+5)+1(-4)|`
`=(1)/(2)|4-16-4|`
= 8 sq. units