First slide

Definition of a circle

Question

$Number of values of c for which set \{(x, y) : x^{2} + y^{2} + 2 x \leq 1\} \cap {(x, y) : x - y + c \geq 0}$ contains only one element

Difficult

A

One
B

Two
C

Zero
D

More than two

Solution

$x^{2} + y^{2} + 2 x \leq 1$
$\Rightarrow {(x + 1)}^{2} + y^{2} \leq {(\sqrt{2})}^{2}$ $I t r e p r e s e n t s i n t e r i o r a n d b o u n d a r y o f t h e c i r c l e h a v i n g c e n t r e a t (- 1, 0) a n d r a d i u s \sqrt{2}$
since the given two regions have only one point common $\Rightarrow$ the line x - y +c =0 is tangent of the circle.
The perpendicular distance from centre to the line = radius

$\frac{|- 1 + c|}{\sqrt{2}} = \sqrt{2}$
$\Rightarrow c = 3; c = - 1$
Reject $c = 3$ ( since number of common points are infinity ). If $c = - 1 \Rightarrow$ number of common points is =1 .

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