Question

If the length of the tangents from P(1, -1) and Q(3, 3) to a circle are $\sqrt{2}\mathrm{and}\sqrt{6}$ then the length of a tangent from R(-1, -5) to the same circle is .

• A. $\sqrt{37}$
• B. $\sqrt{38}$
• C. $\sqrt{35}$
• D. $\sqrt{36}$

Answer 1

The correct option is B $\sqrt{38}$

Let the circle be $x^2+y^2+2\mathrm{gx}+2\mathrm{fy}+c=0$.
Let the tangents from the points P, Q and R touch the circle at A,B and C respectively.
A.T.Q:
${\mathrm{PA}}^2=2$
or 1+1+2g-2f+c=2
or 2g-2f+c=0 ...(i)
and ${\mathrm{PB}}^2=6$
or 9+9+6g+6f+c=6
or 6g+6f+c=-12 ...(ii)
From (i) and (ii):
$g=-1-\frac{c}{3},f=-1+\frac{c}{6}\mathrm{Now}{\mathrm{RC}}^2=1+25-2g-10f+c=26+2+\frac{2c}{3}+10-\frac{5c}{3}+c=38\Rightarrow \mathrm{RC}=\sqrt{38}$