Question

If a chord of the circle $x^2+y^2-4x-2y-c=0$ is trisected at the points $(\frac{1}{3},\frac{1}{3})$ and $(\frac{8}{3},\frac{8}{3}),$ then

• A. Length of the chord $=7\sqrt{2}$
• B. $c=20$
• C. radius of the circle $=25$
• D. $c=25$

Answer 1

Answer: (A), (B), (C)

The correct options are
A Length of the chord $=7\sqrt{2}$
B $c=20$
C radius of the circle $=25$

Equation of the line joining the given points is $y=x$

and the distance between them is $\sqrt{(\frac{7}{3}{)}^2+(\frac{7}{3}{)}^2}=\frac{7\sqrt{2}}{3}.$

If $PQ$ is the chord of the given circle which is trisected at these points

then the equation of $PQ$ is $y=x$

and the length of this chord is $3\times \frac{7\sqrt{2}}{3}=7\sqrt{2}$ .

Let $C(2,1)$ be the center of the given circle and $CL$ be the perpendicular from $C$ to $PQ$,

then the radius of the given circle is $\sqrt{P{L}^2+C{L}^2}$

$\Rightarrow (2{)}^2+(1{)}^2+c=(\frac{7}{\sqrt{2}}{)}^2+(\frac{1}{\sqrt{2}}{)}^2=25$

$\Rightarrow c=20$