Question

Equation of the hyperbola passing through the point $(1,-1)$ and having asymptotes $x+2y+3=0$ and $3x+4y+5=0$ is :

• A. $3x^2-10xy+8y^2-14x+22y+15=0$
• B. $3x^2+10xy+8y^2-14x+22y+35=0$
• C. $3x^2+10xy+8y^2+14x+22y-8=0$
• D. $3x^2+10xy+8y^2+14x+22y+7=0$

Answer 1

The correct option is D $3x^2+10xy+8y^2+14x+22y+7=0$

Pair of Asymptotes: $(x+2y+3)(3x+4y+5)=0$
$\Rightarrow 3x^2+8y^2+10xy+22y+14x+15=0$
We know that equation of hyperbola and (Pair of asymptotes) differs by a constant.
So required family of hyperbola is
$3x^2+8y^2+10xy+22y+14x+15+\lambda =0$
Since hyperbola passes through $(1,-1)$
$\Rightarrow 3(1{)}^2+8(-1{)}^2+10\left(1\right)(-1)+22(-1)+14\left(1\right)+15+\lambda =0$
$\Rightarrow 3+8+(-10)-22+14+15+\lambda =0$
$\Rightarrow \lambda =-8$

Putting the value of $\lambda$ in Equation, we get required hyperbola equation as
$3x^2+10xy+8y^2+14x+22y+7=0$