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Parabola question

2021 · 31 Aug · Shift 1 · Q32
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  5. /2021 · 31 Aug · Shift 1 · Q32

Parabola question

2021 · 31 Aug · Shift 1 · Q32

JEE MainMathematicsParabolaMCQ+4 / −1
The length of the latus rectum of a parabola, whose vertex and focus are on the positive x-axis at a distance R and S (> R) respectively from the origin, is :
  1. A
    4(S + R)
  2. B
    2(S −-− R)
  3. C
    4(S −-− R)
  4. D
    2(S + R)
View written solutionFree

Correct answer: C

  1. Identify the vertex and focus

    Since both lie on the positive xxx-axis:

    • Vertex V=(R,0)V = (R,0)V=(R,0)
    • Focus F=(S,0)F = (S,0)F=(S,0), where S>RS>RS>R

  2. Use the standard form of a parabola

    A parabola with vertex (h,k)(h,k)(h,k) and axis along the positive xxx-direction has equation

    (y−k)2=4a(x−h)(y-k)^2 = 4a(x-h)(y−k)2=4a(x−h)

    where its focus is

    (h+a,k)(h+a,k)(h+a,k)

  3. Compare with the given data

    Here,

    h=R,k=0h=R, \quad k=0h=R,k=0

    and focus is (S,0)(S,0)(S,0).

    So,

    h+a=Sh+a = Sh+a=S R+a=SR+a=SR+a=S a=S−Ra=S-Ra=S−R

  4. Find the length of the latus rectum

    For the parabola

    (y−k)2=4a(x−h),(y-k)^2 = 4a(x-h),(y−k)2=4a(x−h),

    the length of the latus rectum is

    4a4a4a

    Therefore,

    Length of latus rectum=4(S−R)\text{Length of latus rectum} = 4(S-R)Length of latus rectum=4(S−R)

  5. Match with the options

    4(S−R)4(S-R)4(S−R)

    corresponds to Option C.

Final Answer: C. 4(S−R)4(S-R)4(S−R)

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