Circles
Chord Bisected by a Line — Range of Parameter
nta_pyq_2023_jan
Grade 11
Question:
The set of all values of $a^2$ for which the line $x+y=0$ bisects two distinct chords drawn from a point $P\!\left(\dfrac{1+a}{2},\dfrac{1-a}{2}\right)$ on the circle $2x^2+2y^2-(1+a)x-(1-a)y=0$ is equal to:
Step-by-Step Solution
Key Concept: Circle: $x^2+y^2-\frac{1+a}{2}x-\frac{1-a}{2}y=0$. Midpoint of chord on $x+y=0$ has $y=-x$, so $(t,-t)$. Chord through $P$ with midpoint $(t,-t)$: $T=S_1$.
$a^2\in(8,\infty)$.
Correct Answer: 1