CUET PG 2023 — Mathematics PYQ

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Question

CUET PG 2023 — Mathematics PYQ

CUET PG | Mathematics | 2023

If the curve 𝑎𝑦+𝑥²=7 and 𝑥³=𝑦, cut orthogonally at (1, 1), then the value of a is -

Choose the correct answer:

Explanation

Step 1: Pehli curve ka slope ( $m_1$ ) nikalna

Curve ki equation hai:

ay + x^2 = 7

$x$ ke respect mein differentiate karne par:

a \frac{dy}{dx} + 2x = 0

\frac{dy}{dx} = -\frac{2x}{a}

Point $(1, 1)$ par slope $m_1$ :

m_1 = -\frac{2(1)}{a} = -\frac{2}{a}

Step 2: Dusri curve ka slope ( $m_2$ ) nikalna

Curve ki equation hai:

x^3 = y \implies y = x^3

Differentiate karne par:

\frac{dy}{dx} = 3x^2

Point $(1, 1)$ par slope $m_2$ :

m_2 = 3(1)^2 = 3

Step 3: Orthogonality ki condition apply karna

Do curves orthogonally cut karti hain agar:

m_1 \cdot m_2 = -1

Values substitute karte hain:

\left( -\frac{2}{a} \right) \cdot (3) = -1

-\frac{6}{a} = -1

Dono taraf se minus sign cancel karne par:

\frac{6}{a} = 1

a = 6

a = 6