The focus of the parabola $x^{2} + y^{2} - 2 x y - 4 (x + y - 1) = 0$ is

Answer (Detailed Solution Below)View answer ▾

Answer: Option A — (1, 1)

Detailed Solution

$x^{2} + y^{2} - 2 x y = (x - y)^{2}$ , so the curve is $(x - y)^{2} = 4 (x + y - 1)$ .

The lines $x - y = 0$ and $x + y - 1 = 0$ are perpendicular. With $X = \frac{x - y}{2}$ and $Y = \frac{x + y - 1}{2}$ : $2 X^{2} = 42 Y$ , i.e. $X^{2} = 22 Y$ , so $4 a = 22$ and $a = \frac{1}{2}$ .

The axis is $y = x$ and the vertex is $(\frac{1}{2}, \frac{1}{2})$ . The focus is on $y = x$ with $Y = a$ : $x + y - 1 = 1$ , so the focus is $(1, 1)$ .

✓ Check

The directrix is

Y = - a

, i.e.

x + y = 0

. The point

(0, 2)

is on the curve; its distance to

(1, 1)

2

and to

x + y = 0

\frac{2}{2} = 2

⚠ Common trap

(D)

(0, 2)

lies on the parabola (an end of the latus rectum), so it passes a substitution test, but it is not the focus.

Quick tip

$(x - y)^{2} = 4 (x + y - 1)$ : axis $y = x$ , vertex $(\frac{1}{2}, \frac{1}{2})$ , $a = \frac{1}{2}$ , focus $(1, 1)$

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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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