Question

Through a point $(h,k)$ secants are drawn to the circle $(x^2+y^2)=r^2$. The mid-points of the corresponding chords describe the curve

• A. $x^2+y^2+hx+ky=0$
• B. $x^2+y^2-hx-ky=0$
• C. $x^2+y^2-2hx-2ky=0$
• D. $x^2+y^2+2hx+2ky=0$

Answer 1

The correct option is B $x^2+y^2-hx-ky=0$

In given circle, secants are drawn from point $P$ to circle $x^2+y^2=r^2$

Coordinates of midpoint $M(x_1,y_1)$

In $△OMP$

$(OM{)}^2+(MP{)}^2=\left(OP{)}^2\right(x_1^2+y_1^2)+(h-x_1{)}^2+(k-y_1{)}^2=h^2+k^{2}2x_1^2+2y_1^2-2h{x}_1-2k{y}_1=0x_1^2+y_1^2-h{x}_1-k{y}_1=0x^2+y^2-hx-ky=0$