Question

A pair of tangents are drawn from the origin to the circle x^2+y^2+20(x+y)+20=0 the equation of pair of tangents is?

Answer 1

Answer:

The equation of pair of tangents is $2x^{2}+2y^{2}+5xy=0$

Step-by-step explanation:

Given that a pair of tangents are drawn from the origin to the circle $x^2+y^2+20(x+y)+20=0$

we have to find the the equation of pair of tangents

Comparing given equation $x^2+y^2+20(x+y)+20=0$ with $x^{2} + y^{2} + 2gx + 2fy + c = 0$ we get center and radius will be

Center:(-f,-g)=(-10,-10)

Radius:$\sqrt{f^{2}+g^{2}-c}=\sqrt{100+100-20}=\sqrt{180}$

Equation of line passing through origin is y=mx ⇒ mx-y=0 which is the tangent to that of origin.

Perpendicular distance=radius= $|\frac{-10m+10}{\sqrt{1+m^{2}}}|=\sqrt{180}$

Solving above equation we get

$-10m+10=\pm\sqrt{180}\sqrt{1+m^{2}}$

Squaring we get

$(-10m+10)^{2}=180+180m^{2}$

⇒ $m=\frac{-1}{2}, m=-2$

∴ the two equations are

$y=-2x$ and $y=\frac{-1}{2}x$

⇒ 2x+y=0 and  x+2y=0

Combined equations are

(2x+y)(x+2y)=0

⇒ $2x^{2}+2y^{2}+5xy=0$

Hence, the equation of pair of tangents is $2x^{2}+2y^{2}+5xy=0$