Question

Equation of the tangent at the point (3, -1) to the ellipse 2x² + 9y² = 3 is

1. 2x - 3y - 3 = 0
2. 2x + 3y - 3 = 0
3. 2x + y - 3 = 0
4. None of these

Answer 1

Correct Answer - Option 4 : None of these

Concept:

The equation of the tangent to an ellipse \(\rm \dfrac {x^2}{a^2}+ \dfrac {y^2}{b^2} = 1\) at the point (x₁, y₁) is \(\rm \dfrac {xx_1}{a^2}+ \dfrac {yy_1}{b^2} = 1\)

Calculations:

Given, the point is (3, -1) = (x₁, y₁) and equation of ellipse is 2x² + 9y² = 3

⇒ \(\rm \dfrac {x^2}{\frac 32}+ \dfrac {y^2}{\frac 13} = 1\)

The equation of the tangent to an ellipse \(\rm \dfrac {x^2}{a^2}+ \dfrac {y^2}{b^2} = 1\) at the point (x₁, y₁) is \(\rm \dfrac {xx_1}{a^2}+ \dfrac {yy_1}{b^2} = 1\)

Here, x1 = 3, y₁ = -1 \(\rm a^2 = \dfrac 3 2 \) and \(\rm b^2 = 1\)

Equation of the tangent from the point (3, -1) to the ellipse 2x² + 9y² = 3 is

⇒ \(\rm \dfrac {3x}{\frac 3 2}- \dfrac {y}{\frac 1 3 } = 1\)

⇒ 2x - 3y - 1 = 0

Hence, Equation of the tangent from the point (3, -1) to the ellipse 2x² + 9y² = 3 is 2x - 3y - 1 = 0