Maths Q12 – JEE-MAIN 2026

Let the parabola

y = x^{2} + p x + q

passing through the point

(1, - 1)

be such that the distance between its vertex and the

x

-axis is minimum. Then the value of

p^{2} + q^{2}

is:

Clue 1 of 3: Vertex of a Parabola

Recall the formula for the coordinates of the vertex of a parabola $y = a x^{2} + b x + c$ .

🧠 Full Solution Path

Step 1: Determine Vertex Coordinates and Constraint✦ Active

The given parabola is $y = x^{2} + p x + q$ . The coordinates of its vertex are $V (- \frac{p}{2}, q - \frac{p^{2}}{4})$ . Since the parabola passes through the point $(1, - 1)$ , we substitute these coordinates into the equation:

- 1 = (1)^{2} + p (1) + q ⟹ - 1 = 1 + p + q ⟹ p + q = - 2

Step 2: Express Vertex y-coordinate in terms of one variable○ Expand

From the constraint $p + q = - 2$ , we can express $q$ as $q = - 2 - p$ . Now, substitute this into the $y$ -coordinate of the vertex, $y_{v} = q - \frac{p^{2}}{4}$ :

y_{v} = (- 2 - p) - \frac{p^{2}}{4} = - \frac{p^{2}}{4} - p - 2

This can be rewritten as $y_{v} = - \frac{1}{4} (p^{2} + 4 p + 8)$ .

Step 3: Minimize Distance and Calculate Result○ Expand

The distance between the vertex and the $x$ -axis is $| y_{v} |$ . We need to minimize this distance. The expression for $y_{v}$ is a quadratic in $p$ that opens downwards. Its maximum value occurs at $p = - \frac{4}{2 (1)} = - 2$ . At this value of $p$ :

y_{v} = - \frac{1}{4} ((- 2)^{2} + 4 (- 2) + 8) = - \frac{1}{4} (4 - 8 + 8) = - \frac{1}{4} (4) = - 1

Since $y_{v} = - 1$ is the maximum value of $y_{v}$ , and $y_{v}$ is always less than or equal to $- 1$ , the minimum value of $| y_{v} |$ is $| - 1 | = 1$ . This occurs when $p = - 2$ . Now, use $p = - 2$ in the relation $p + q = - 2$ to find $q$ :

- 2 + q = - 2 ⟹ q = 0

Finally, calculate $p^{2} + q^{2}$ :

p^{2} + q^{2} = (- 2)^{2} + (0)^{2} = 4 + 0 = 4

💡 Teacher's Secret Hint

Remember that for a parabola $y = a x^{2} + b x + c$ with $a > 0$ , the vertex is the minimum point. Thus, $y_{v}$ is the minimum value of $y$ . To minimize $| y_{v} |$ , we need $y_{v}$ to be as close to zero as possible.

✦ STEM Console utilizes AI models to generate step-by-step explanations and math clues. AI can make mistakes.

Maths Question 12 – JEE-MAIN 2026

Clue 1 of 3: Vertex of a Parabola

🧠 Full Solution Path