Question

In a hyperbola portion of tangent intercepted between asymptotes is bisected at the point of contact. Consider a hyperbola whose centre is at origin. A line $x+y=2$ touches this hyperbola at $P(1,1)$ and intersects the asymtotes at $A$ and $B$ such that $AB=6\sqrt{2}$ units.

Equation of asymptotes are

• A. $5xy+2x^2+2y^2=0$
• B. $3x^2+4y^2+6xy=0$
• C. $2x^2+2y^2-5xy=0$
• D. none of these

Answer 1

The correct option is A $5xy+2x^2+2y^2=0$

The equation of tangent in parametric form is given by
$\frac{x-1}{-\frac{1}{\sqrt{2}}}=\frac{y-1}{\frac{1}{\sqrt{2}}}=\pm 3\sqrt{2}$
or $A\equiv (4,-2),B\equiv (-2,4)$
The equations of asymptotes ($OA$ and $OB$) are given by
$y+2=\frac{-2}{4}(x-4)$
or $2y+x=0$
and $y-4=\frac{4}{-2}(x+2)$
or $2x+y=0$
Hence, the combined equation of asymptotes is
$(2x+y)(x+2y)=0$
or $2x^2+2y^2+5xy=0$