GATE 2025 EE GA Q.3: solution and answer

GATE 2025 · EEGA Q.3 · 1 mark · MCQ

The relationship between two variables $x$ and $y$ is given by $x + p y + q = 0$ and is shown in the figure. Find the values of $p$ and $q$ .

Note: The figure shown is representative.

GATE 2025 EE GA question 3: A Cartesian graph on a square grid. — Figure from the GATE 2025 EE paper. It is part of the question.

(A)
$p = - \frac{1}{2}; q = 2$
(B)
$p = 2; q = - 2$
(C)
$p = \frac{1}{2}; q = 4$
(D)
$p = 2; q = 4$

Answer and solution

ANSWER

$p = - \frac{1}{2}$ and $q = 2$ , so the answer is option (A).

Official answer: ASource: Official answer key, GATE 2025 EE (IIT Roorkee)

Read two points off the graph. The line meets the $x$ -axis at $(- 2, 0)$ and the $y$ -axis at $(0, 4)$ . It also passes through $(3, 10)$ , which we use later as a check.
Slope from the two intercepts: $m = \frac{4 - 0}{0 - ( - 2 )} = 2$ . The $y$ -intercept is $c = 4$ .
Slope-intercept form $y = m x + c$ gives $y = 2 x + 4$ .
Rearrange into the given form, where the coefficient of $x$ must be 1: $2 x - y + 4 = 0$ . Divide by 2 to get $x - \frac{1}{2} y + 2 = 0$ .
Compare with $x + p y + q = 0$ : $p = - \frac{1}{2}$ and $q = 2$ . The coefficient of $x$ is fixed at 1, so these values are unique.
Check with $(- 2, 0)$ : $- 2 + 0 + 2 = 0$ . Check with $(3, 10)$ : $3 - \frac{1}{2} (10) + 2 = 3 - 5 + 2 = 0$ . Both points satisfy the equation, so the answer is (A).

KEY IDEA

Put the two intercepts straight into $x + p y + q = 0$ . At $(- 2, 0)$ the equation becomes $- 2 + q = 0$ , so $q = 2$ . At $(0, 4)$ it becomes $4 p + q = 0$ , so $p = - \frac{1}{2}$ . The coefficient of $x$ is fixed at 1, so the equation cannot be rescaled and $p$ and $q$ are unique.

WATCH OUT FOR

Seeing slope 2 and intercept 4 on the graph and choosing (D) $p = 2; q = 4$ . Those numbers belong to $y = 2 x + 4$ , not to the form $x + p y + q = 0$ . A second trap is a sign slip when solving for $x$ . From $y = 2 x + 4$ you get $x = \frac{1}{2} y - 2$ , which is $x - \frac{1}{2} y + 2 = 0$ , not $x + \frac{1}{2} y + \dots$ . Getting that sign wrong points you to (C). Always check the answer by substituting one plotted point, such as $(- 2, 0)$ .

From the GATE 2025 EE paper. Every question of this paper · Download the full paper · All 22 Geometry questions, with notes

GATE 2025 EE GA Q.3: solution and answer

Answer and solution

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