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Parabola question

2008 · Shift 0 · Q44
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Parabola question

2008 · Shift 0 · Q44

JEE MainMathematicsParabolaMCQ+4 / −1
A parabola has the origin as its focus and the line x=2x=2x=2 as the directrix. Then the vertex of the parabola is at :
  1. A
    (0,2)(0,2)(0,2)
  2. B
    (1,0)(1,0)(1,0)
  3. C
    (0,1)(0,1)(0,1)
  4. D
    (2,0)(2,0)(2,0)
View written solutionFree

Correct answer: B

  1. Given data

    • Focus of the parabola: origin F=(0,0)F=(0,0)F=(0,0)
    • Directrix: line x=2x=2x=2

  2. Key fact about a parabola The vertex lies midway between the focus and the directrix, measured along the axis perpendicular to the directrix.

  3. Find the axis Since the directrix is the vertical line x=2x=2x=2, the axis of the parabola is horizontal. It must pass through the focus (0,0)(0,0)(0,0), so the axis is: y=0y=0y=0

  4. Find the foot of the perpendicular from the focus to the directrix From (0,0)(0,0)(0,0) to the line x=2x=2x=2, the perpendicular meets the directrix at: (2,0)(2,0)(2,0)

  5. Vertex is the midpoint The vertex is the midpoint of (0,0)(0,0)(0,0) and (2,0)(2,0)(2,0): (0+22,0+02)=(1,0)\left(\frac{0+2}{2},\frac{0+0}{2}\right)=(1,0)(20+2​,20+0​)=(1,0)

  6. Match with options (1,0)\boxed{(1,0)}(1,0)​ So the correct option is B.

  7. Comparison with stored answer Stored correct answer: B

    Our derived answer is also B, so they agree.

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