GATE 2026 IN Q.53: solution and answer

GATE 2026 · INQ.53 · 2 marks · MSQ

Given $f (x, y) = x^{2} - 2 x y + y^{2}$

The complete contour of the equation $f (x, y) = 1$ is described by the option(s) ____.

(A)
A line, $x - y = 1$
(B)
A circle with radius 1
(C)
An ellipse with length of major axis equal to 1
(D)
A line, $x - y = - 1$

See the question in the paper

Tap to open full size

Answer and solution

ANSWER

(A) and (D). The contour $f (x, y) = 1$ is $(x - y)^{2} = 1$ , which is the pair of parallel lines $x - y = 1$ and $x - y = - 1$ .

Official answer: A, DSource: Official answer key, GATE 2026 IN (IIT Guwahati)

Spot the perfect square: $x^{2} - 2 x y + y^{2} = (x - y)^{2}$ . The contour $f (x, y) = 1$ is therefore $(x - y)^{2} = 1$ .
Take square roots and keep both signs: $x - y = \pm 1$ . The contour is the pair of parallel lines $x - y = 1$ and $x - y = - 1$ .
Check with the conic discriminant. For the quadratic part $A x^{2} + B x y + C y^{2}$ , here $A = 1, B = - 2, C = 1$ , so $B^{2} - 4 A C = 4 - 4 = 0$ . That is the parabolic case, and here it is degenerate: the curve splits into two straight lines. A circle or an ellipse needs $B^{2} - 4 A C < 0$ , so options (B) and (C) are ruled out.
Check one point on each line. $(1, 0)$ lies on $x - y = 1$ and gives $f = 1 - 0 + 0 = 1$ . $(0, 1)$ lies on $x - y = - 1$ and gives $f = 0 - 0 + 1 = 1$ . Both lines belong to the contour.
The question asks for the complete contour. That contour is exactly the union of the two lines, so (A) and (D) together describe it, and both are correct.

KEY IDEA

$x^{2} - 2 x y + y^{2}$ is the perfect square $(x - y)^{2}$ , so a level set $(x - y)^{2} = k$ with $k > 0$ is two parallel lines $x - y = \pm k$ , not a closed curve. The discriminant $B^{2} - 4 A C = 0$ gives the same answer: it rules out circles and ellipses.

WATCH OUT FOR

Keeping only the positive root and marking (A) alone. The complete contour is both lines, because $(x - y)^{2} = 1$ also holds when $x - y = - 1$ . Another trap is to see squared terms with equal coefficients and guess a circle. That ignores the $- 2 x y$ cross term, which turns the expression into a perfect square.

From the GATE 2026 IN paper. Every question of this paper · Download the full paper · All 39 Calculus questions, with notes

GATE 2026 IN Q.53: solution and answer

Answer and solution

More Calculus questions