Question

The equation of the locus of the mid points of the chord of the circle $4x^2+4y^2-12x+4y+1=0$ that subtends an angle of $\frac{2\pi }{3}$ at its centre is

• A. $x^2+y^2-3x+y+\frac{16}{31}=0$
• B. $x^2+y^2-3x+y-\frac{31}{16}=0$
• C. $x^2+y^2+3x+y+\frac{31}{16}=0$
• D. $x^2+y^2-3x+y+\frac{31}{16}=0$

Answer 1

The correct option is D $x^2+y^2-3x+y+\frac{31}{16}=0$

For the circle $4x^2+4y^2-12x+4y+1=0$
Radius $=3/2$
Let $E(x,y)$ is mid-point of chord $AB$
So, in $\Delta OAE$,
$\frac{3}{2}cos{60}^{\circ }=\sqrt{(\frac{3}{2}-x{)}^2+(y+\frac{1}{2}{)}^2}$
${\left(\frac{3}{4}\right)}^2={\left(\frac{3}{2}\right)}^2+x^2-3x+y^2+{\left(\frac{1}{2}\right)}^2+y$
$x^2+y^2-3x+y+\frac{10}{4}-\frac{9}{16}=0$
$\Rightarrow x^2+y^2-3x+y+\frac{31}{16}=0$