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Maths-

General

Easy

Question

If the tangent to the parabola y² = ax makes an angle of 45º with x-axis, then the point of contact is

$(\frac{a}{2}, \frac{a}{2})$
$(\frac{a}{4}, \frac{a}{4})$
$(\frac{a}{2}, \frac{a}{4})$
$(\frac{a}{4}, \frac{a}{2})$

The correct answer is: $open parentheses fraction numerator a over denominator 4 end fraction comma fraction numerator a over denominator 2 end fraction close parentheses$

y² = ax $rightwards double arrow$ 2y $fraction numerator d y over denominator d x end fraction equals a$
Slope of tangent $fraction numerator d y over denominator d x end fraction$ = $fraction numerator a over denominator 2 y end fraction$ … (i)
Slope of given tangent m = tan 45º = 1…(ii)
From (i) and (ii) are same $therefore fraction numerator a over denominator 2 y end fraction equals 1$ $rightwards double arrow$ y = $fraction numerator a over denominator 2 end fraction$
and x = a/4 $rightwards double arrow$ (a/4, a/2)

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