First slide

Hyperbola in conic sections

Question

$E : \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ and $P : y^{2} = 4 b x, a > b .$
Statement 1: The tangent at the positive end of the
minor axis of the ellipse. E passes through the positive
end of the latus rectum of the parabola $P .$
Statement 2: If the latus rectum of the parabola $P$ is
same as that of the ellipse $E,$ then eccentricity of $E$
is $1 / \sqrt{2}$

Moderate

A

STATEMENT- I is True, STATEMENT-2 is True; S TATEMENT-2 is a correct explanation for STATEMENT- I
B

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

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

STATEMENT-I is False, STATEMENT-2 is True

Solution

Tangent at the positive end of the minor axis of $E$ is
$y = b$ which meets the parabola $y^{2} = 4 b x$ at the point
$(\frac{b}{4}, b)$ which is not an end of the latus rectum of $P .$
so statement- I is false.
In statement-2 if $x = b$ and $x = a e$ represent the same
line then $b = a e$
$\begin{array}{l} \Rightarrow \frac{b}{a} = e \Rightarrow a^{2} (1 - e^{2}) = a^{2} e^{2} \\ \Rightarrow e^{2} = \frac{1}{2} \Rightarrow e = \frac{1}{\sqrt{2}} . \end{array}$
Thus statement-2 is true.

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