Question

The distance between the foci of the conic $7x^2-9y^2=63$is equal to

• A. $8$
• B. $4$
• C. $3$
• D. $7$
• E. $12$

Answer 1

The correct option is A

$8$

Explanation for the correct option:

Find the distance between foci:

Let us simplify the given equation as

$$\begin{gathered} 7x^2-9y^2=63 \\ \Rightarrow \frac{7x^2}{63}-\frac{9y^2}{63}=1 \\ \Rightarrow \frac{x^2}{9}-\frac{y^2}{7}=1\to \left(i\right) \end{gathered}$$

Equating $\left(i\right)$ with the equation of the hyperbola:

$$\begin{gathered} \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 \\ \Rightarrow a^2=9\text{and}{b}^2=7\left[\text{from}\left(i\right)\right] \\ \Rightarrow a=3\text{and}b=\sqrt{7} \end{gathered}$$

We know that,

$$\begin{gathered} e=\sqrt{1+\frac{b^2}{a^2}} \\ =\sqrt{1+\frac{7}{9}} \\ =\frac{4}{3} \end{gathered}$$

We know that,

The distance between the foci is equal to $2ae$

$$\begin{gathered} =2\times 3\times \frac{4}{3}\left[\because a=3;e=\frac{4}{3}\right] \\ =8 \end{gathered}$$

Hence, option (A) is the correct answer.