First slide

Hyperbola in conic sections

Question

Statement-1: Equation of a circle on the ends of a
latus rectum of the hyperbola $\frac{x^{2}}{16} - \frac{y^{2}}{9} = 1$ as a diameter is $16 x^{2} + 16 y^{2} \pm 160 x + 319 = 0$
Statement-2: Focus of the parabola $y^{2} = 20 x$ coin cides with a focus of the hyperbola $\frac{x^{2}}{16} - \frac{y^{2}}{9} = 1$

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

$Foci (\pm 5, 0),$ length of the latus rectum $= \frac{2 \times 9}{4}$
So the equation of a required circle is $(x \pm 5)^{2} + y^{2} =$ ${(\frac{9}{4})}^{2}$
$\begin{array}{l} \Rightarrow 16 (x^{2} + y^{2} \pm 10 x + 25) = 81 \\ or 16 x^{2} + 16 y^{2} \pm 160 x + 319 = 0 \end{array}$
and statement-1 is true, statement-2 is also true as the focus of $y^{2} = 20 x is (5, 0)$ but does not justify statement-1.

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