Question

The locus of the mid-point of the focal chord of the parabola y2 = 4x is a parabola, whose vertex is
A) (0,0) B) (1,0) C) (0,1) D) (1,1)
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Answer 1

Answer:

(0,0)

Step-by-step explanation:

y2=4x

(0*2) =(4*0)

Answer 2

Answer:

(B)The vertex of the parabola is (1,0)

Step-by-step explanation:

The given parabola is

${y}^{2} = 4x$

Comparing it with the standard equation of parabola

${y}^{2} = 4ax = > a = 1$

We know that the ends of focal chords of a parabola are

$(at^{2},2at) \: and \: ( \frac{a}{ {t}^{2} } , - \frac{2a}{t} )$

Substituting a=1,The end points become

$(t^{2},2t) \: and \: ( \frac{1} { {t}^{2} } , \frac{ - 2}{t} )$

Let (X,Y) be the midpoint of the focal chord

$= > Y = \frac{2t - \frac{2}{t} }{2} = t - \frac{1}{t} \\ = > X = \frac{ {t}^{2} + \frac{1}{ {t}^{2} } }{2} \\ = > 2X = (t - \frac{1}{t} ) {}^ 2 + 2 = Y {}^{2} + 2 \\ = > Y {}^{2} = 2(X - 1)$

Therefore,from above equation the vertex of the parabola is (1,0)

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