Question

The length of the tangent drawn from the point $(2,3)$ to the circle $2(x^2+y^2)-7x+9y-11=0$.

• A. $18$
• B. $14$
• C. $\sqrt{14}$
• D. $\sqrt{28}$

Answer 1

The correct option is C $\sqrt{14}$

REF.Image

$x^2+y^2-\frac{7x}{2}+\frac{9y}{2}-\frac{11}{2}=0$

$A\equiv (\frac{7}{4},\frac{-9}{4})$

Radius = $\sqrt{\frac{49+81}{16}+\frac{11}{2}}=\sqrt{\frac{65}{8}+\frac{44}{8}}$

$=\sqrt{\frac{109}{8}}$

$AC=\sqrt{(\frac{2-7}{4}{)}^2+(3+\frac{9}{4}{)}^2}$

$\sqrt{\frac{1}{16}+\frac{441}{16}}=\sqrt{\frac{221}{8}}$

$BC=\sqrt{(AC{)}^2-(AB{)}^2}$

$=\sqrt{\frac{221-109}{8}}=\sqrt{\frac{112}{8}}$

$=\sqrt{14}$ units