Question

For each point (x,y) on an ellipse, the sum of the distances from (x,y) to the points (2,0) and (-2,0) is 8. Then the positive value of x so that (x,3) lies on the ellipse is ___

• A. 2
• B. $2\sqrt{3}$
• C. $\frac{1}{\sqrt{3}}$
• D. 4

Answer 1

The correct option is A

2

Given that the sum of the distances from (x,y) to the points (2,0) and (-2,0) is 8.
Let the equation of the ellipse be $\frac{X^2}{a^2}+\frac{Y^2}{b^2}=1$
$\therefore 2a=8\Rightarrow a=4;c=2b^2=a^2-c^2\Rightarrow b^2=16-4=12$

The equation of the ellipse is $\frac{X^2}{16}+\frac{Y^2}{12}=1$

If (x,3) lies on the ellipse, then,
$\frac{x^2}{16}+\frac{3^2}{12}=1\Rightarrow \frac{x^2}{16}=\frac{1}{4}\Rightarrow x^2=4x=2\text{[only positive square root is considered]}$