Question

An ellipse has its centre at $(1,-1)$ and semi major axis equal to $8$. If this ellipse passes through the point $(1,3)$, its eccentricity is equal to

• A. $\frac{1}{\sqrt{2}}$
• B. $\frac{1}{2}$
• C. $\frac{\sqrt{3}}{2}$
• D. none of these

Answer 1

The correct option is C $\frac{\sqrt{3}}{2}$

Given center of ellipse is at (1,-1)

passes through = (1,3)

length of semi major axis = 8

Let equation of ellipse with center at (1,-1) &

a = 8 is

$\frac{(x-1{)}^2}{6^4}+\frac{(y+1{)}^2}{b^2}=1----\left(1\right)$

eq (1) passes through

$(\frac{1-1}{6^4}{)}^2+(\frac{(3+1)}{b^2}{)}^2=1$

$b^2=(4{)}^2=16$

[b = 4]

now eccentricity ellipse $e=\sqrt{\frac{a^2-b^2}{a^2}}=\sqrt{1-\frac{b^2}{a^2}}$

$e=\sqrt{1-\frac{16}{64}}=\sqrt{\frac{48}{64}}=\frac{2\sqrt{12}}{8}=\sqrt{\frac{3}{4}}$