1.

Find the equation of ellipse whose latus rectum is half the major axis and focus is at (3,0).

Answer»

Let the major axis is 2a.

Focus is (3,0) lies on x-axis.

Hence, Major axis of ellipse lies on x-axis.

Given that   \(\frac{2b^2}{a}=\frac{2a}{2} ⇒\frac{2b^2}{a}=a\)

⇒ \(\)2b2 = a2 ⇒  \(b^2=\frac{a^2}{2}\)     ----(i)

∴  \(e =\sqrt{1-\frac{b^2}{a^2}}\) \(= \sqrt{1 - \frac{1}{2}}\)  (From (i))

 = \(e =\frac{1}{√2}\)

∴ Focus of ellipse is (ae,o) = (3,0)

⇒ ae = 3

⇒ \(a×\frac{1}{√2}=3\)

⇒ \(a=3√2\)

∴ b2 = \(\frac{18}{2}\) = 9   (from (ii))

b = 3

Hence , equation of ellipse is 

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

⇒ \(\frac{x^2}{18}+\frac{y^2}{9}=1\)

 is equation of required ellipse.


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