Question

The locus of the mid points of the chords of the circle $x^2 + y^2 - ax - by = 0$ which subtend a right angle at (a/2, b/2) is-

• A. ax + by = 0
• B. ax + by = $a^2 + b^2$
• C. $x^2 + y^2 - ax - by + \frac{a^2+b^2}{8}=0$
• D. $x^2 + y^2 - ax - by - \frac{a^2+b^2}{8} = 0$

Answer 1

Answer: (C)

$x^2 + y^2 - ax - by + \frac{a^2+b^2}{8}=0$

Answer 2

The given circle is $x^2 + y^2 - ax - by = 0$. Its center is $O_1 = (a/2, b/2)$ and its radius squared is $r_1^2 = (a/2)^2 + (b/2)^2 = \frac{a^2+b^2}{4}$. Let $P = (a/2, b/2)$ be the point where the chords subtend a right angle. Notice that $P$ is the center of the circle, i.e., $P \equiv O_1$. Let $M(h, k)$ be the midpoint of a chord $AB$ of the circle. The line segment $O_1M$ is perpendicular to the chord $AB$. The condition is that the chord $AB$ subtends a right angle at $P$. Since $P=O_1$, this means $\angle AO_1B = 90^\circ$. In the isosceles triangle $\triangle AO_1B$ (with $O_1A = O_1B = r_1$), the line segment $O_1M$ bisects the angle $\angle AO_1B$. Thus, $\angle AO_1M = \angle BO_1M = 90^\circ/2 = 45^\circ$. In the right-angled triangle $\triangle AO_1M$ (with $\angle AMO_1 = 90^\circ$), we have: $O_1M = O_1A \cos(\angle AO_1M)$ $O_1M = r_1 \cos(45^\circ) = r_1 \frac{1}{\sqrt{2}}$ Squaring both sides: $O_1M^2 = \frac{r_1^2}{2}$ Substituting $r_1^2 = \frac{a^2+b^2}{4}$: $O_1M^2 = \frac{1}{2} \left(\frac{a^2+b^2}{4}\right) = \frac{a^2+b^2}{8}$ The distance $O_1M$ is the distance between $M(h, k)$ and $O_1(a/2, b/2)$: $O_1M^2 = (h - a/2)^2 + (k - b/2)^2$ Equating the two expressions for $O_1M^2$: $(h - a/2)^2 + (k - b/2)^2 = \frac{a^2+b^2}{8}$ Expanding this equation: $h^2 - ah + \frac{a^2}{4} + k^2 - bk + \frac{b^2}{4} = \frac{a^2+b^2}{8}$ $h^2 + k^2 - ah - bk + \frac{a^2+b^2}{4} = \frac{a^2+b^2}{8}$ $h^2 + k^2 - ah - bk = \frac{a^2+b^2}{8} - \frac{a^2+b^2}{4}$ $h^2 + k^2 - ah - bk = -\frac{a^2+b^2}{8}$ $h^2 + k^2 - ah - bk + \frac{a^2+b^2}{8} = 0$ Replacing $(h, k)$ with $(x, y)$ to get the locus equation: $x^2 + y^2 - ax - by + \frac{a^2+b^2}{8} = 0$