Question

What is the length of tangent from a point $P(x_1,y_1)$ to the circle $x^2+y^2+2gx+2fy+c=0.$

• A.
• B.
• C.
• D.

Answer 1

The correct option is B

Let O be the centre of the circle and T be tha point at which tangent is drawn.

$△PTO$ is right angled triangle.

Circles equation is,

$x^2+y^2+2gh+2fy+c=0$

$(x+g{)}^2+(y+f{)}^2-g^2-f^2+c=0$

$i.e.,(x+g{)}^2+(y+f{)}^2=g^2+f^2-c$

$\therefore$ Comparing with the standard form.

Centre $\equiv (-g,-f)$

Radius$\equiv \sqrt{g^2+f^2-c}$

Since $△PTO$ is right angled

$P{T}^2=P{O}^2-O{T}^2$

$={\sqrt{(x_1+g{)}^2+(y_1+f{)}^2}}^2-{\sqrt{g^2+f^2-c}}^2$

$=(x_1+g{)}^2+(y^1+f{)}^2-(g^2+f_2-c)$

$=x_1^2+y_1^2+2g{x}_1+2f{y}_1+c$

$\therefore PT=\sqrt{x_1^2+y_1^2+2g{x}_1+2f{y}_1+c}$