Question

If a chord of the circle $x^2+y^2-4x-2y-c=0$ is trisected at the points (1/3, 1/3) and (8/3, 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
$\sqrt{(7/3{)}^2+(7/3{)}^2}i.e.,7\sqrt{2}/3.$


If PQ is the chord of the given circle which is trisected at these points then 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 centre of the given circle and CL be the perpendicular from C to PQ, then the radius of the given circle is
$\sqrt{P{L}^2+C{L}^2}=(7/\sqrt{2}{)}^2+(1/\sqrt{2}{)}^2=25$

$\Rightarrow (2{)}^2+(1{)}^2+c=25$
$\Rightarrow c=20.$