The eccentricity of the conic represented by 4x² + 9y² = 36:-

Concept:

• An ellipse is defined as the set of points in a plane in which the sum of distances from two fixed points is constant.
• The eccentricity of the ellipse is less than 1, i.e. e < 1.

• The general equation of an ellipse is written as:

  \(\frac{x^2}{a^2}+\frac{y^2}{b^2} =1\) and the eccentricity formula is written as \(e = \sqrt{1-\frac{b^2 }{a^2 }} \)

Calculations:

Consider 4x2 + 9y2 = 36

divide the whole equation by 36, we get

\(\frac{x^2}{3^2}+\frac{y^2}{2^2} =1\)

Thus,  the eccentricity formula is written as \(e = \sqrt{1-\frac{b^2 }{a^2 }} \)

Since a = 3 and b = 2, we have

\(e = \sqrt{1-\frac{2^2 }{3^2 }} \)

⇒ \(e = \sqrt{1-\frac{4 }{9 }} \)

⇒ \(e = \sqrt{\frac{5 }{9 }} \)

⇒ \(e =\frac{ \sqrt{5 }}{3 }\)

Hence, the correct answer is option 2).