NDA 1 2025 Maths, question 58

Topic: Analytical geometry

Q58

A tangent to the parabola $y^{2} = 4x$ is inclined at an angle $45^{\circ}$ with the positive direction of $x$ -axis. What is the point of contact of the tangent and the parabola?

(a) $(1,1)$
(b) $\left( 2,2\sqrt{2} \right)$
(c) $\left( 1/2,1/\sqrt{2} \right)$
(d) $(1,2)$ Correct answer

Solution.

Answer: (d) $(1,2)$

The required tangent slope is $\tan 45^{\circ} = 1$ .
Differentiating $y^{2} = 4x$ gives $dy/dx = 2/y$ .
Setting $2/y = 1$ gives $y = 2$ , and then $x = y^{2}/4 = 1$ .
The point of contact is $(1,2)$ .