A normal to the curve $x^{2} = 4 y$ passes through the point (1, 2). The distance of the origin from the normal is

Answer (Detailed Solution Below)View answer ▾

Answer: Option D — $\frac{3}{2}$

Detailed Solution

At $(2 t, t^{2})$ the tangent slope is $\frac{x}{2} = t$ , so the normal is $x + t y = 2 t + t^{3}$ .

Through (1, 2): $1 + 2 t = 2 t + t^{3}$ , so $t^{3} = 1$ and $t = 1$ .

Normal $x + y = 3$ ; distance from the origin $= \frac{3}{2}$ .

✓ Check

(1, 2) satisfies

1 + 2 = 3

, and the foot point (2, 1) lies on

x^{2} = 4 y

⚠ Common trap

(C)

\frac{1}{2}

is the distance of the tangent

x - y - 1 = 0

at (2, 1), not the normal.

Quick tip

Normal to $x^{2} = 4 y$ at $(2 t, t^{2})$ : $x + t y = 2 t + t^{3}$ .

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Parabola — Preparation

This question tests your understanding of Parabola, a key topic in the Quantitative Aptitude section. The detailed solution above uses a step-by-step approach to show the reasoning, and where available, a shortcut that helps you solve similar questions in under 90 seconds during the actual exam.

A parabola is the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix), so its eccentricity is exactly 1. The standard form y^2 = 4ax has its vertex at the origin, focus at (a, 0), directrix x = -a, and latus rectum of length 4a. The other standard forms open left, up or down. Exam questions test identifying the focus, directrix, axis and latus rectum, the parametric point (at^2, 2at), focal distances, and equations of tangents and normals. A key result is that the line y = mx + c touches y^2 = 4ax when c = a/m. Drawing a quick sketch of which way the parabola opens prevents most sign errors. Practise the parabola questions below, each with a step-by-step solution.

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