Question

If the coordinates of the vertices of a hyperbola are $(9,2)$ and $(1,2)$ and the distance between its foci is $10$ units. Then

• A. length of its latus rectum is $9$ units
• B. length of its latus rectum is $\frac{9}{2}$ units
• C. equation of the hyperbola is $\frac{(x-5{)}^2}{9}$- \dfrac{(y-2)^2}{16}=1$
• D. equation of the hyperbola is $\frac{(x-5{)}^2}{16}$- \dfrac{(y-2)^2}{9}=1$

Answer 1

Answer: (B), (D)

equation of the hyperbola is $\frac{(x-5{)}^2}{16}$- \dfrac{(y-2)^2}{9}=1$

Since, ordinates of the vertices are equal, thus transverse axis is parallel to $x-$ axis and conjugate axis is parallel to $y-$ axis.
Mid point of vertices $=$ Center of hyperbola $=(5,2)$
Length of transverse axis $=$ Distance between $(9,2)$ and $(1,2)=8$
$\Rightarrow 2a=8$
$\Rightarrow a=4$
Distance between its two foci$=2ae=10$
$\Rightarrow ae=5$
Now, $b^2=a^2(e^2-1)=9$
Length of latus rectum$=2\cdot \frac{b^2}{a}$
Length of latus rectum $=\frac{9}{2}$ units
Equation of hyperbola is
$\frac{(x-5{)}^2}{16}$$-\frac{(y-2{)}^2}{9}=1$