A parabola with axis parallel to $x$ axis passes through $(0, 0), (2, 1), (4, - 1) .$ Its length of latus rectum is

\frac{2}{3}

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\frac{1}{4}

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\frac{7}{3}

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\frac{1}{3}

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Solution

The correct option is D $\frac{1}{3}$
$(y - k)^{2} = 4 a (x - h)$
It passes through $(0, 0)$
$k^{2} = - 4 a h$ ---(1)
It passes through $(2, 1)$
$(k - 1)^{2} = 4 a (2 - h)$ ---(2)
It passes through $(4, - 1)$
$(k + 1)^{2} = 4 a (4 - h)$ ---(3)
(2) + (3)
$2 (k^{2} + 1) = 4 a (6 - 2 h)$
$2 k^{2} + 2 = 24 a + 2 k^{2}$ ( form eq (1) )
$4 a = \frac{1}{3}$
$∴$ length of LR $= \frac{1}{3}$

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