Consider the parabola whose focus is at (0,0) and tangent at vertex is x−y+1=0. Then
A
the length of latus rectum is 2√2
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B
the length of the chord of parabola on the x-axis is 4√2
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C
equation of directrix is x−y−2=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 4√2 The distance between the focus and the tangent at the vertex =|0−0+1|√12+12=1√2
The directrix is the line parallel to the tangent at vertex and at a distance 2×1√2 from the focus.
Let equation of directrix be x−y+λ=0,
where λ√12+12=2√2 ⇒λ=2 ⇒ Equation of directrix is x−y+2=0
Let P(x,y) be any moving point on the parabola.
Then OP=PM ⇒x2+y2=(x−y+2√12+12)2 ⇒2x2+2y2=(x−y+2)2 ⇒x2+y2+2xy−4x+4y−4=0...(1)
Latus rectum length =2× (distance of focus from directrix) =2∣∣
∣∣0−0+2√12+12∣∣
∣∣ =2√2
Now, solving parabola with x-axis by putting y=0 in equation (1). x2−4x−4=0 ⇒x=4±√322=2±2√2 ⇒ Length of chord on x-axis is 4√2.
Since, the chord 3x+2y=0 passes through the focus, so it is focal chord.
Hence, tangents at the extremities of chord are perpendicular.