A focus of an ellipse having eccentricity 12 is at 0, 0 and a directrix is the line x=4. Then the equation of one such ellipse is

TS EAMCET · Maths · Ellipse

A focus of an ellipse having eccentricity $\frac{1}{2}$ is at $(0, 0)$ and a directrix is the line $x = 4 .$ Then the equation of one such ellipse is

A $\frac{9 x^{2}}{64} + \frac{3 y^{2}}{16} = 1$
B $\frac{(2 x + 1)^{2}}{32} + \frac{y^{2}}{16} = 1$
C $\frac{(3 x + 4)^{2}}{64} + \frac{y^{2}}{32} = 1$
D $(3 x + 4)^{2} + 12 y^{2} = 64$

Verified Solution

Answer & Solution

Correct Answer

(D) $(3 x + 4)^{2} + 12 y^{2} = 64$

Step-by-step Solution

Detailed explanation

A focus of an ellipse is at origin. Directrix is x=4 and eccentricity is e=12 Major axis is along x-axis , ⇒ae-ae=4 ⇒a12-a2=4 ⇒a=83 By the help of options we get the idea that this case is of a>b Then we can say that e=1-b2a2 ⇒14=1-9b264⇒b2=163…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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