The locus of the mid-point of the line segment joining the focus ot a moving point on the parabola $y^{2}$ = 4ax is another parabola with directrix

x = - a

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$x = - \frac{a}{2}$

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x = 0

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$x = \frac{a}{2}$

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Solution

The correct option is C
x = 0

$S = (a, 0), P (a t^{2}, 2 a t) . I f (h, k) b e m i d - p o i n t, t h e n 2 h = a (1 + t^{2}), 2 k = 2 a t o r k = a t$
$∴ 2 x = a (\frac{1 + \frac{y^{2}}{a^{2}}}{}) o r y^{2} = 2 a (\frac{x - \frac{a}{2}}{}) o r Y^{2} = 4 A X$
$∴ D i r e c t r i x i s X = - A o r x - \frac{a}{2} = - \frac{2 a}{4} o r x = 0.$

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The locus of the mid-point of the line segment joining the focus ot a moving point on the parabola y2 = 4ax is another parabola with directrix

The correct option is C x = 0 S=(a,0),P(at2,2at).If(h,k) be mid−point, then 2h=a(1+t2),2k=2at or k=at ∴ 2x =a (1+y2a2 )or y2=2a(x−a2 )or Y2=4AX ∴ Directrix is X = −A or x −a2=−2a4 or x = 0.

The locus of the mid-point of the line segment joining the focus ot a moving point on the parabola $y^{2}$ = 4ax is another parabola with directrix