Question

If $OA$ and $OB$ are two equal chords of the circle $x^2+y^2-2x+4y=0$ perpendicular to each other and passing through the origin $O$, the slopes of $OA$ and $OB$ are the roots of the equation

• A. $3m^2+8m-3=0$
• B. $3m^2-8m-3=0$
• C. $8m^2-3m-8=0$
• D. $8m^2+3m-8=0$

Answer 1

The correct option is B $3m^2-8m-3=0$

Let the equations of $OA$ and $OB$ be $y-mx=0$ and $my+x=0$
Since $OA=OB$ lengths of the perpendiculars from the center.
$(1,-2)$ of the circle on $OA$ and $OB$ are also equal.

$\Rightarrow \left|\frac{-2-m}{\sqrt{1+m^2}}\right|=\left|\frac{-2m+1}{\sqrt{1+m^2}}\right|\Rightarrow (m+2{)}^2=(2m-1{)}^2$

$\Rightarrow 3m^2-8m-3=0$