Maths Q20 – JEE-MAIN 2026

Let

\frac{x^{2}}{f (a^{2} + 7 a + 3)} + \frac{y^{2}}{f (3 a + 15)} = 1

represent an ellipse with major axis along

y

-axis, where

f

is a strictly decreasing positive function on

R

. If the set of all possible values of

a

R - [α, β]

, then

α^{2} + β^{2}

is equal to :

Clue 1 of 3: Ellipse Properties

Recall the conditions for an ellipse and how the major axis orientation relates to the denominators in the standard equation.

🧠 Full Solution Path

Step 1: Identify conditions for the ellipse and major axis✦ Active

The given equation is $\frac{x^{2}}{f (a^{2} + 7 a + 3)} + \frac{y^{2}}{f (3 a + 15)} = 1$ . For this to represent an ellipse, the denominators must be positive. Since $f$ is a positive function, $f (a^{2} + 7 a + 3) > 0$ and $f (3 a + 15) > 0$ are always satisfied. The major axis is along the $y$ -axis, which implies that the denominator of the $y^{2}$ term must be greater than the denominator of the $x^{2}$ term.

f (3 a + 15) > f (a^{2} + 7 a + 3)

Step 2: Apply the property of a strictly decreasing function○ Expand

Given that $f$ is a strictly decreasing function, if $f (x_{1}) > f (x_{2})$ , then it must be that $x_{1} < x_{2}$ . Applying this property to the inequality from Step 1:

3 a + 15 < a^{2} + 7 a + 3

Rearranging the terms to form a quadratic inequality:

a^{2} + 7 a + 3 - 3 a - 15 > 0

a^{2} + 4 a - 12 > 0

💡 Teacher's Secret Hint

Remember that the inequality sign flips when dealing with a strictly decreasing function.

Step 3: Solve the quadratic inequality and calculate

α^{2} + β^{2}

○ Expand

Factor the quadratic expression:

(a + 6) (a - 2) > 0

This inequality holds when $a < - 6$ or $a > 2$ . The set of all possible values of $a$ is $(- \infty, - 6) \cup (2, \infty)$ . This can be expressed as $R - [- 6, 2]$ . Comparing this with the given form $R - [α, β]$ , we identify $α = - 6$ and $β = 2$ . Now, calculate $α^{2} + β^{2}$ :

α^{2} + β^{2} = (- 6)^{2} + (2)^{2} = 36 + 4 = 40

💡 Teacher's Secret Hint

Ensure correct identification of $α$ and $β$ from the interval notation.

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Maths Question 20 – JEE-MAIN 2026

Clue 1 of 3: Ellipse Properties

🧠 Full Solution Path