Circles
Circle
Allen Star Batch
Grade 11
Question:
Chords of the circle $x^2 + y^2 = 9$ are drawn such that segments intercepted from the chords by the curve $y^2 - 4x - 4y = 0$ subtend right angle at the origin. If the locus of the middle points of the chords with respect to circle is a curve $S$, then:
$S$ is a pair of line
$S$ is a circle
$S$ passes through the origin
$S$ meets the given circle $x^2 + y^2 = 9$ at $A$ and $B$ and the tangents at $A$ and $B$ to the circle $x^2 + y^2 = 9$ intersect at $(4, 4)$
Step-by-Step Solution
Key Concept: The locus of chord midpoints satisfying a perpendicularity condition is found by homogenization and eliminating the parameter.
Given a chord with midpoint $P(h,k)$, the chord equation is $hx + ky = h^2 + k^2$. Homogenizing $y^2 - 4x - 4y = 0$ using this relation yields a pair of perpendicular lines when $4h + 4k - h^2 - k^2 = 0$. The locus of $P$ is $x^2 + y^2 - 4x - 4y = 0$, and the chord of contact from $(4,4)$ is $4x + 4y = 9$.
Correct Answer: 2,3,4