CUET PG 2023 — Mathematics PYQ

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Question

CUET PG 2023 — Mathematics PYQ

CUET PG | Mathematics | 2023

The tangent to the hyperbola 𝑥²−𝑦²=3 are parallel to the straight line 2𝑥+𝑦+8=0 at the following points:

Choose the correct answer:

Explanation

1. Identify the Slope

The given straight line is:

2x + y + 8 = 0 \implies y = -2x - 8

The slope ( $m$ ) of this line is $-2$ .

Since the tangents are parallel to this line, the slope of the tangents must also be:

m = -2

2. Differentiate the Hyperbola Equation

The equation of the hyperbola is:

x^2 - y^2 = 3

Differentiating both sides with respect to $x$ :

2x - 2y\frac{dy}{dx} = 0

\frac{dy}{dx} = \frac{x}{y}

The derivative $\frac{dy}{dx}$ represents the slope of the tangent at any point $(x, y)$ .

3. Find the Relation between $x$ and $y$

Equating the derivative to the required slope $m = -2$ :

\frac{x}{y} = -2 \implies x = -2y

4. Substitute back into the Hyperbola Equation

Substitute $x = -2y$ into the original equation $x^2 - y^2 = 3$ :

(-2y)^2 - y^2 = 3

4y^2 - y^2 = 3

3y^2 = 3

y^2 = 1 \implies y = \pm 1

5. Determine the Points

Case 1: If $y = 1$

$x = -2(1) = -2$

Point: $(-2, 1)$
Case 2: If $y = -1$

$x = -2(-1) = 2$

Point: $(2, -1)$

Final Answer:

The points of tangency are $(-2, 1)$ and $(2, -1)$ .