Question

If the centre, vertex and focus of a hyperbola are $(4,3),(8,3),(10,3)$ respectively, then the equation of the hyperbola is

• A. $\frac{(x-4{)}^2}{16}-\frac{(y-3{)}^2}{24}=1$
• B. $\frac{(x-4{)}^2}{16}-\frac{(y-3{)}^2}{20}=1$
• C. $\frac{(x-4{)}^2}{20}-\frac{(y-3{)}^2}{16}=1$
• D. $\frac{(x-4{)}^2}{20}-\frac{(y-3{)}^2}{12}=1$

Answer 1

The correct option is B $\frac{(x-4{)}^2}{16}-\frac{(y-3{)}^2}{20}=1$

Given ,
Centre $=(4,3)$
Vertex $=(8,3)$
Focus $=(10,3)$
So, Hyperbola will have transverse axis parallel to $x$-axis
Now $a=$ distance between centre and vertex
$\Rightarrow a=8-4=4$
and $ae=$ distance between centre and focus
$ae=10-4=6$
$\Rightarrow e=\frac{3}{2}$
$\Rightarrow b^2=a^2(e^2-1)=20$
Thus equation of hyperbola is $\frac{(x-4{)}^2}{16}-\frac{(y-3{)}^2}{20}=1$