If the area of the auxiliary circle of the ellipse x²/a² + y²/b² = 1 (a > b) is twice the area of the ellipse, then the eccentricity of the ellipse is

• a. 1/√3
• b. 1/2
• c. 1/√2
• d. √3/2

The problem involves an ellipse defined by the equation x²/a² + y²/b² = 1, where a > b. To find the eccentricity of the ellipse given that the area of its auxiliary circle is twice the area of the ellipse, we need to first calculate the areas involved.

Area of the Ellipse

The area of the ellipse can be calculated using the formula:

Area of Ellipse = πab

Area of the Auxiliary Circle

The auxiliary circle of the ellipse has a radius equal to the semi-major axis 'a'. Therefore, its area is given by:

Area of Auxiliary Circle = πa²

Setting Up the Equation

According to the problem, the area of the auxiliary circle is twice the area of the ellipse. This gives us the equation:

πa² = 2(πab)

Simplifying the Equation

We can simplify this equation:

• Cancel π from both sides: a² = 2ab
• Rearranging gives: a² - 2ab = 0
• Factoring out a: a(a - 2b) = 0

Since a cannot be zero, we have:

a - 2b = 0 or a = 2b

Finding the Eccentricity

The eccentricity 'e' of the ellipse is calculated using the formula:

e = √(1 - (b²/a²))

Substituting a = 2b into the eccentricity formula:

e = √(1 - (b²/(2b)²)) = √(1 - (b²/4b²)) = √(1 - 1/4) = √(3/4) = √3/2

Final Answer

The eccentricity of the ellipse is √3/2, which corresponds to option d.