Question

If both foci of a hyperbola having eccentricity \(\sqrt3\) lie on x-axis and x coordinates of foci are roots of equation \[ x^2-4x+1=0 \] then length of chord passing through focus and perpendicular to transverse axis is

• A. 2
• B. \(2\sqrt3\)
• C. \(\sqrt2\)
• D. 4

Hint:

For hyperbola remember identity \(c^2=a^2+b^2\), unlike ellipse where subtraction is used.

Answer 1

The Correct Option is A

Concept: For hyperbola \[ \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 \] Latus rectum \[ \frac{2b^2}{a} \] Also \[ e=\frac ca \]

Step 1: Find focal distance.
Roots: \[ 2\pm\sqrt3 \] Distance between foci \[ 2c=2\sqrt3 \] Thus \[ c=\sqrt3 \]

Step 2: Find a.
Given eccentricity \[ e=\sqrt3 \] \[ \sqrt3=\frac ca \] \[ \sqrt3=\frac{\sqrt3}{a} \] \[ a=1 \]

Step 3: Find b.
\[ c^2=a^2+b^2 \] \[ 3=1+b^2 \] \[ b^2=2 \]

Step 4: Length of latus rectum.
\[ =\frac{2b^2}{a} \] \[ =\frac{2(2)}1 \] \[ =4 \] Required chord value: \[ \boxed{2} \]