Circles
Circle
Allen Star Batch
Grade 11
Question:
A circle touches the line $x + y - 2 = 0$ at $(1,1)$ and cuts the circle $x^2 + y^2 + 4x + 5y - 6 = 0$ at $P$ and $Q$. Then:
$PQ$ can never be parallel to the given line $x + y - 2 = 0$
$PQ$ can never be perpendicular to the given line $x + y - 2 = 0$
$PQ$ always passes through $(6, -4)$
$PQ$ always passes through $(-6, 4)$
Step-by-Step Solution
Key Concept: The locus of an external point whose chord of contact is tangent to a second circle is found by applying the tangency condition (distance from center equals radius).
The chord of contact $PQ$ from external point $(h,k)$ to circle $x^2 + y^2 = a^2$ has equation $hx + ky = a^2$. For $PQ$ to be tangent to circle $(x-a)^2 + y^2 = a^2$, the distance from center $(a,0)$ to line $hx + ky = a^2$ must equal radius $a$. This gives $\frac{|ah-a^2|}{\sqrt{h^2+k^2}} = a$, leading to $(h-a)^2 = h^2 + k^2$. Expanding yields the locus $y^2 = a(a-2x)$.
Correct Answer: 1,3