A point P in the first quadrant lies on $y^{2} = 4 a x$ , $a > 0$ , and is at a distance $5 a$ from its focus. Which of the following points lies on the locus of P?

Answer (Detailed Solution Below)View answer ▾

Answer: Option B — (1, 1)

Detailed Solution

The focal distance of a point on $y^{2} = 4 a x$ is $x + a$ . So $x + a = 5 a$ and $x = 4 a$ .

$y^{2} = 4 a \cdot 4 a = 16 a^{2}$ , so $y = 4 a$ (first quadrant) and $P = (4 a, 4 a)$ .

As $a$ varies, P moves on the line $y = x$ . Of the options, only $(1, 1)$ lies on it.

✓ Check

a = 1

P = (4, 4)

, focus

(1, 0)

, distance

9 + 16 = 5 = 5 a

⚠ Common trap

(A)

(1, 0)

is the focus when

a = 1

, not a position of P.

Quick tip

Focal distance $x + a = 5 a \Rightarrow P = (4 a, 4 a)$ , so the locus is $y = x$

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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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A point P in the first quadrant lies on $y^{2} = 4 a x$ , $a > 0$ , and is at a distance $5 a$ from its focus. Which of the following points lies on the locus of P?

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A point P in the first quadrant lies on y2=4ax, a>0, and is at a distance 5a from its focus. Which of the following points lies on the locus of P?

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Parabola — Preparation

A point P in the first quadrant lies on $y^{2} = 4 a x$ , $a > 0$ , and is at a distance $5 a$ from its focus. Which of the following points lies on the locus of P?