First slide

Equation of line in Straight lines

Question

The maximum value of $y = \sqrt{(x - 3)^{2} + {(x^{2} - 2)}^{2}} - \sqrt{x^{2} + {(x^{2} - 1)}^{2}} is$

Difficult

A

3
B

$\sqrt{10}$
C

$2 \sqrt{5}$
D

None of these

Solution

$y = f (x) = \sqrt{{(x^{2} - 2)}^{2} + (x - 3)^{2}} - \sqrt{x^{2} + {(x^{2} - 1)}^{2}}$
$Note that the first term describes the distance between P (x, x^{2}) and A (3, 2) where as the second term describes the$
$distance between P (x, x^{2}) and B (0, 1) .$
$Now P A - P B \leq A B for possible positions of P$
$Hence {[f (x)]}_{max} = distance between A B = \sqrt{9 + 1} = \sqrt{10}$

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