First slide

Hyperbola in conic sections

Question

Statement-1: If the foci of a hyperbola are at the points (4, 1) and (–6, 1), eccentricity is 5/4 then the length of the transverse axis is 8.
Statement-2: Distance between the foci of a hyper bola is equal to the product of its eccentricity and the length of the transverse axis

Moderate

A

STATEMENT-1 is True, STATEMENT-2 is True;
STATEMENT-2 is a correct explanation for STATEMENT-1
B

STATEMENT-1 is True, STATEMENT-2 is True;
STATEMENT-2 is NOT a correct explanation for STATEMENT-1
C

STATEMENT-1 is True, STATEMENT-2 is False
D

STATEMENT-1 is False, STATEMENT-2 is True

Solution

Statement-2 is true, as the distance between the foci of $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 is 2 a e where 2 a$ is the length of the transverse axis and $e$ is the eccentricity
Using it in statement-1, length of the transverse axis is $\frac{\sqrt{(4 + 6)^{2}}}{5 / 4} = \frac{10 \times 4}{5} = 8$
and thus statement-1 is also true.

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