Question

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

Answer 1

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

T=0

$3x-9\times 2y=9$

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

Equation of the hyperpola is $x^2-9y^2=9$ ..........(2)

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

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

$(3+6y{)}^2-9y^2=9$

$(1+2y{)}^2-y^2=1$

$1+4y^2+4y-y^2=1$

$3y^2+4y=0$

y(3y+4)=0

$y=0$ & $y=-\frac{4}{3}$

$\Rightarrow$ When y=0,x=3

when $y=-\frac{4}{3},x=6\left(\frac{-4}{3}\right)+3$

$x=-8+3=-5$

point of intersection being (3,0) & $(-5,-\frac{4}{3})$

Area of triangle formed by points (3,2),(3,0) & $(-5,-\frac{4}{3})$ is

$=\frac{1}{2}\left|\begin{array}{ccc}1& 1& 1\\ 3& 3& -5\\ 2& 0& -\frac{4}{3}\end{array}\right|$