Circles
Circle
star_batch_jee_advanced_2025
Grade 11
Question:
MATCH THE FOLLOWING:
(A) If the point $(c, c+2)$ is an interior point of smaller segment of the curve $x^2 + y^2 - 4 = 0$ made by the chord of the curve whose equation is $3x + 4y + 12 = 0$, then the value of $c$ is
(B) If the set represented by $\{(x, y) | x^2 + y^2 + 2x \leq 1\} \cap \{(x, y) | x - y + c \geq 0\}$ contains only one point then the value of $c$ is
(C) $ABCD$ is a square of unit area. If a circle touches two sides of $ABCD$ and passes through exactly one of its vertices. Then the radius of this circle is
(D) If tangents are drawn from $(4, 4)$ to the circle $x^2 + y^2 - 2x - 2y - 7 = 0$ to meet the circle at $A$ and $B$, then length of the chord $AB$ is
Step-by-Step Solution
Key Concept: Distance from center to line equals radius for tangency; use parametric representations for chord-angle relationships.
For option (B), a line is tangent to a circle when $\frac{|1-c|}{\sqrt{2}} = \sqrt{2}$, giving $c = -1$. For option (C), the circle $x^2 + y^2 - 2rx - 2ry + r^2 = 0$ passes through $(1,1)$, so $r^2 - 4r + 2 = 0$, yielding $r = 2 - \sqrt{2}$ (since $r < 1$). For option (D), with $\sin\theta = \frac{1}{\sqrt{2}}$, we get $\theta = \frac{\pi}{4}$ and $AB = 2r\cos\theta = 3\sqrt{2}$.
Correct Answer: [A-q] [B-p][C-t] [D-r]