If two distinct chords of a parabola $y^{2} = a x$ passing through the point $(a, a)$ are bisected by the line $x + y = 1,$ then the length of the latus rectum cannot be :

3

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4

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5

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6

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Solution

The correct options are
C $5$
D $6$
Let at any point $(t, 1 - t)$ on the line $x + y = 1$ , the two distinct chords are bisected.
$T = S_{1}$
$\Rightarrow y (1 - t) - \frac{a (x + t)}{2} = (1 - t)^{2} - a t \Rightarrow 2 (1 - t) y - a (x + t) = 2 (t - 1)^{2} - 2 a t$
This chord passes through $(a, a)$
$\Rightarrow 2 a (1 - t) - a (a + t) = 2 (t - 1)^{2} - 2 a t$
$\Rightarrow 2 a - a^{2} - a t = 2 t^{2} - 4 t + 2 \Rightarrow 2 t^{2} + (a - 4) t + a^{2} - 2 a + 2 = 0$
$t$ must have two distinct value for two distinct chord
$△ > 0 \Rightarrow (a - 4)^{2} - 8 (a^{2} - 2 a + 2) > 0 \Rightarrow - 7 a^{2} + 8 a > 0 \Rightarrow a (7 a - 8) < 0 \Rightarrow a \in (0, \frac{8}{7}) \Rightarrow 4 a \in (0, \frac{32}{7})$

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If two distinct chords of a parabola y2=ax passing through the point (a,a) are bisected by the line x+y=1, then the length of the latus rectum cannot be :

If two distinct chords of a parabola $y^{2} = a x$ passing through the point $(a, a)$ are bisected by the line $x + y = 1,$ then the length of the latus rectum cannot be :