Question

Tangents are drawn from $(4,4)$ to the circle $x^2+y^2-2x-2y-7=0$ to meet the circle at $A$ and $B$. What is the length of the chord $AB$?

Answer 1

The general equation of the circle is $x^2+y^2+2gx+2fy+c=0$

The given equation of the circle is $x^2+y^2-2x-2y-7=0$

$\therefore g=-1,f=-1,c=-7$

Length of the tangent to the circle, $\mathit{L}\mathbf{=}\sqrt{{{\mathbf{x}}_{\mathbf{1}}}^{\mathbf{2}}\mathbf{+}{{\mathbf{y}}_{\mathbf{1}}}^{\mathbf{2}}\mathbf{+}\mathbf{2}\mathbf{g}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}\mathbf{2}\mathbf{f}{\mathbf{y}}_{\mathbf{1}}\mathbf{+}\mathbf{c}}$ where $\left(x_1,y_1\right)=\left(4,4\right)$

$=\sqrt{{\left(4\right)}^2+{\left(4\right)}^2+2\times -1\times 4+2\times -1\times 4-7}$

$=\sqrt{16+16-8-8-7}$

$=\sqrt{9}=3$

Radius of the circle, $\mathit{R}\mathbf{=}\sqrt{{\mathbf{g}}^{\mathbf{2}}\mathbf{+}{\mathbf{f}}^{\mathbf{2}}\mathbf{-}\mathbf{c}}$

$=\sqrt{{\left(-1\right)}^2+{\left(-1\right)}^2-\left(-7\right)}$

$=\sqrt{1+1+7}=\sqrt{9}=3$

Length of the chord $AB$$\mathbf{=}\frac{\mathbf{2}\mathbf{L}\mathbf{R}}{\sqrt{{\mathbf{L}}^{\mathbf{2}}\mathbf{+}{\mathbf{R}}^{\mathbf{2}}}}$

$=\frac{2\times 3\times 3}{\sqrt{{\left(3\right)}^2+{\left(3\right)}^2}}$

$=\frac{18}{\sqrt{9+9}}$

$=\frac{18}{\sqrt{18}}$

$=\frac{18}{\sqrt{18}}\times \frac{\sqrt{18}}{\sqrt{18}}$

$=\frac{18\sqrt{18}}{18}=\sqrt{18}=3\sqrt{2}$

Hence, the length of the chord $AB=3\sqrt{2}units$