CircleHardBloom L3
Question
The value of ' $c$ ' for which the set, $\left\{ (x,y) \mid x^{2} + y^{2} + 2x \leq 1 \right\} \cap \{(x,y) \mid x - y + c \geq 0\}$ contains only one point in common is:
Options
A.$( - \infty, - 1\rbrack \cup \lbrack 3,\infty)$
B.$\{ - 1,3\}$
C.$\{ - 3\}$
D.$\{ - 1\}$
Solution
[Key Concept: The set intersection is non-empty when the line meets the disc — bound $c$ by the distance of the line from the centre.]
Line is tangent to circle as shown
$$\Rightarrow \frac{1 - c}{\sqrt{2}} = \sqrt{2} $$$\lbrack\because( - 1,0)$ lies above the line
$${y - x - c = 0\rbrack }{\Rightarrow \ c = - 1}$$
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