Question

Through a fixed point $(h,k)$ secants are drawn to the circle $x^2+y^2=r^2$ Then the locus of the midpoints of the chords intercepted by the circle is

• A. $x^2+y^2=hx+ky$
• B. $x^2-y^2=hx+ky$
• C. $x^2+y^2=hx-ky$
• D. $x^2-y^2=hx-ky$

Answer 1

The correct option is A $x^2+y^2=hx+ky$

If $P(x_1,y_1)$ is the midpoint of the chord, we get the line
$S_1=S_{11}$,
$x_{1}x+y_{1}y-r^2=x_1^2+y_1^2-r^2$
It passes through $(h,k)\therefore h{x}_1+k{y}_1=x_1^2+y_1^2$
$\therefore$ The locus of $P$ is $x_1^2+y_1^2=hx+ky$
Replace $x_1\to x$ and $y_1\to y$ we get
$x^2+y^2=hx+ky$