Question

If $OA$ and $OB$ are the tangents to the circle $x^2+y^2-6x-8y+21=0$ drawn from the origin $O$, then $AB$ equals

• A. $\frac{17}{3}$
• B. $\frac{4}{5}\sqrt{21}$
• C. $11$
• D. None of these

Answer 1

The correct option is B $\frac{4}{5}\sqrt{21}$

equation of circle $\Rightarrow x^2+y^2-6x-8y+21=0$

radius $\Delta =\sqrt{9+16-21}=2$

$AB$ is chord of contact & its equation is

$x.x_1+y{y}_1+9(x+x_1)+f(y+y_1)+c=0$

$(x_1,y_1)=(0,0)$

$0+0-3(x+0)-4(y+0)+21=0$

$3x+4y-21=0$

Perpendicular distance from $(3,4)$ to line $l_1$

$CM=\frac{3\left(3\right)+4\left(4\right)-21}{\sqrt{9+16}}=\frac{4}{5}$

$AM=\sqrt{A{C}^2-C{H}^2}=\sqrt{4-\frac{16}{25}}=\frac{2}{5}\sqrt{21}$

$AB=2AM=\frac{4}{5}\sqrt{21}$

option $B$ is correct.