Question

If $x=9$ is the chord of contact of the hyperbola $x^2-y^2=9$, then the equation of the corresponding pair of tangents is

• A. $9x^2-8y^2+18x-9=0$
• B. $9x^2-8y^2-18x+9=0$
• C. $9x^2-8y^2-18x-9=0$
• D. $9x^2-8y^2+18x+9=0$

Answer 1

Answer: (B)

Let a pair of tangents be drawn from the point
$\left(x_1,y_1\right)$ to the hyperbola
$x^2-y^2=9$
Then the chord of contact will be
$x{x}_1-y{y}_1=9$ ... (i)
But the given chord of contact is
$x=9$ .... (ii)
As (i) and (ii) represent the same line, these equations should be identical and, hence,
$\frac{x_1}{1}=-\frac{y_1}{0}=\frac{9}{9}$
or $x_1=1,y_1=0$
Therefore, the equation of pair of tangents drawn from $(1,0)$ to $x^2-y^2=9$ is
$\left(x^2-y^2-9\right)\left(1^2-0^2-9\right)=(x\cdot 1-y\cdot 0-9{)}^2\text{ (Using }S{S}_1=T^2)$
or $\left(x^2-y^2-9\right)(-8)=(x-9{)}^2$
or $-8x^2+8y^2+72=x^2-18x+81$
or $9x^2-8y^2-18x+9=0$