First slide

Extreme values and Periodicity of Trigonometric functions

Question

If (x, y) satisfies the equation ${(x + 5)}^{2} + {(y - 12)}^{2} = 196$ then minimum value of $\sqrt{x^{2} + y^{2}}$ is

Moderate

A

0
B

-1
C

1
D

-2

Solution

Let $x + 15 = 14 \cos θ, y - 12 = 14 \sin θ$
$\Rightarrow {(x + 5)}^{2} + {(y - 12)}^{2} = {(14)}^{2}$
Now $x = - 5 + 14 \cos θ$
and $y = 12 + 14 \sin θ$
$\Rightarrow x^{2} + y^{2} = 365 + 28 (12 \sin θ - 5 \cos θ)$
$∴ \min = 365 - 28 \times 13 = 1$

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