First slide

Hyperbola in conic sections

Question

Let $P (x, y)$ be a variable point such that $|\sqrt{(x - 3)^{2} + (y - 2)^{2}} - \sqrt{(x - 6)^{2} + (y + 2)^{2}}| = 3 .$
Statement-1: $P$ traces a hyperbola whose eccentricity is $\frac{5}{3}$ .
Statement-2: $P$ traces a hyperbola such that the equation of its conjugate axis is $6 x - 8 y = 27$

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

Distance of $P$ from the fixed points $F$ 1(3, 2) and $F_{2} (6, - 2)$ is constant 3.
So by definition, locus of P is a hyperbola with foci F1 and F2, length of the transverse axes is 3. If e is the eccentricity then $F_{1} F_{2} = 2 a e where 2 a = 3 .$
$\Rightarrow e = \frac{1}{3} \sqrt{9 + 16} = \frac{5}{3}$ and the statement-1 is true.
n statement-2 slope of the conjugate axis is $- \frac{6 - 3}{- 2 - 2}$ $= \frac{3}{4}$ and it passes through the mid-point of $F_{1} F_{2}$
So its equation is
$y = \frac{3}{4} (x - \frac{9}{2})$
$\Rightarrow 6 x - 8 y = 27$
Thus statement-2 is also true but does not lead to statement-1.

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App