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Q.

Let a variable circle touches a line, which is tangent to the parabola y2=4x at  (4, 4) and variable circle passes through focus of the parabola y2=4x. If locus of its centre is conic ‘C’, then then which of the following is/are true?

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a

Focus of conic is (4, 0)

b

 latus rectum of conic ' C' is 25

c

Directrix of conic ‘C’ is x – 2y + 4 = 0

d

Axis of the conic ‘C’ is 2x + y – 2 = 0

answer is A, C, D.

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Detailed Solution

Equation of tangent to y2=4x at (4,4) is x – 2y + 4 = 0

Suppose that the center of the circle is Px1,y1

The distance from Px1,y1 to the above line is equal to the distance from Px1,y1 to the focus of the parabola

hence, 

x1-2y1+41+4=x1-12+y1-02

The above equation is the parabola having focus at 1,0 and directrix x-2y+4=0

Axis of the conic is a line perpendicular to the above line and passing through the focus

it is 2x+y=2

Hence the length of the latusrectum is =2×55=25

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