Question

The equations of the tangents drawn from the point $(0,1)$ to the circle $x^2+y^2-2x+4y=0$ are

• A. $2x-y+1=0,x+2y-2=0$
• B. $2x-y-1=0,x-2y-2=0$
• C. $2x-y+1=0,x+2y+2=0$
• D. $2x-y-1=0,x+2y+2=0$

Answer 1

The correct option is A $2x-y+1=0,x+2y-2=0$

Let equation of tangent with slope $=m$ and point $(0,1)$

$(y-1)=m(x-0)\Rightarrow y=mx+1$

Intersection point

$x^2+(mx+1{)}^2-2x+4(mx+1)=0(1+m^2)x^2+(-2+6m)x+5=0$

For $y=mx+1$ to be tangent, discriminant $=0$

$(6m-2{)}^2-4\times 5(1+m^2)=0$

$36m^2+4-24m-20m^2+20=0$

$16m^2-20m+24=0\Rightarrow 2m^2-3m-2=0$

$(2m+1)(m-2)=0m=2,\frac{-1}{2}$

Equation of tangents

$y=\frac{-1}{2}x+1\Rightarrow x+2y-2=0$

and

$y=2x+1\Rightarrow 2x-y+1=0$