Question

If the vertices of a hyperbola be at $(-2,0)$ and $(2,0)$ and one of its foci be at $(-3,0),$ then which of the following is/are correct?

• A. eccentricity of the hyperbola is $\frac{3}{2}$
• B. $(6,5\sqrt{2})$ does not lie on hyperbola
• C. length of the latus rectum is $5$ unit
• D. $(2\sqrt{6},5)$ does not lie on hyperbola

Answer 1

Answer: (A), (B), (C)

The correct options are
A eccentricity of the hyperbola is $\frac{3}{2}$
B $(6,5\sqrt{2})$ does not lie on hyperbola
C length of the latus rectum is $5$ unit

Let hyperbola be
$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$
Given $a=2,ae=3\Rightarrow e=\frac{3}{2}$
As, $(ae{)}^2=a^2+b^2$
$\Rightarrow 9=4+b^2$
$\Rightarrow b^2=5$
$\therefore$ Equation of Hyperbola is
$\frac{x^2}{4}-\frac{y^2}{5}=1$
Length of the latus rectum
$=\frac{2b^2}{a}=5$ unit
On substituting the coordinates of the points given in options, we can say that only the point $(6,5\sqrt{2})$ does not satisfy the equation of hyperbola.