Suppose an ellipse and a hyperbola have the same pair of foci on the `x`-axis with centres at the origin and they intersect at `(2,2)`. If the eccentricity of the ellipse is `(1)/(2)`, then the eccentricity of the hyperbola, is

Options

A. `sqrt((7)/(4))`
B. `sqrt((7)/(3))`
C. `sqrt((5)/(4))`
D. `sqrt((5)/(3))`

Explanation

To solve the problem step by step, we will derive the eccentricity of the hyperbola given the eccentricity of the ellipse and their intersection point.

### Step 1: Define the equations of the ellipse and hyperbola

The standard form of the ellipse centered at the origin with foci on the x-axis is given by:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
...