Circles
Circle
Allen Star Batch
Grade 11
Question:
Two chords are drawn from the point $P(h, k)$ on the circle $x^2 + y^2 = hx + ky$. If the $y$-axis divides both the chords in the ratio $2:3$, then which of the following may be correct?
$k^2 > 15h^2$
$15k^2 > h^2$
$h^2 5h^2$
Step-by-Step Solution
Key Concept: The discriminant condition for real solutions of $y$ (or $x$) determines the envelope of a family of circles parametrized by $h$ and $k$.
Substituting $x = \frac{-2h}{3}$ into the family of circles gives the discriminant condition $D > 0 \Rightarrow k^2 - \frac{40}{9}h^2 > 0$. Alternatively, substituting $x = \frac{-3h}{2}$ yields $y^2 - ky + \frac{15h^2}{4} = 0$ with $D > 0 \Rightarrow k^2 - 15h^2 > 0$. These conditions define the locus envelope, representing the boundary where circles cease to exist.
Correct Answer: 1,2,4