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Question

Consider the parabola whose focus is at (0,0) and tangent at vertex is xy+1=0. Then

A
the length of latus rectum is 22
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B
the length of the chord of parabola on the x-axis is 42
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C
equation of directrix is xy2=0
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D
tangents drawn to the parabola at the extremities of the chord 3x+2y=0 intersect at an angle of π3
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Solution

The correct option is B the length of the chord of parabola on the x-axis is 42
The distance between the focus and the tangent at the vertex =|00+1|12+12=12
The directrix is the line parallel to the tangent at vertex and at a distance 2×12 from the focus.
Let equation of directrix be xy+λ=0,
where λ12+12=22
λ=2
Equation of directrix is xy+2=0

Let P(x,y) be any moving point on the parabola.
Then OP=PM
x2+y2=(xy+212+12)2
2x2+2y2=(xy+2)2
x2+y2+2xy4x+4y4=0 ...(1)


Latus rectum length =2× (distance of focus from directrix)
=2∣ ∣00+212+12∣ ∣
=22

Now, solving parabola with x-axis by putting y=0 in equation (1).
x24x4=0
x=4±322=2±22
Length of chord on x-axis is 42.

Since, the chord 3x+2y=0 passes through the focus, so it is focal chord.
Hence, tangents at the extremities of chord are perpendicular.

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