Take timed practice tests on Parabola for JEE Main and JEE Advanced with session-wise drills, score review, and explanation-led revision.
Six 20-question timed sessions plus one 60-question module test. Each question is original and calibrated from the uploaded material pattern without copying PDF wording.
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1. The standard equation of a parabola opening rightward is:
Explanation: Standard parabolas: y² = 4ax (opens right), y² = −4ax (opens left), x² = 4ay (opens up), x² = −4ay (opens down). Focus at (a, 0) for y² = 4ax.
2. For y² = 4ax, the focus is at:
Explanation: For y² = 4ax: vertex at origin, focus at (a, 0), directrix at x = −a, axis = x-axis. Any point on parabola is equidistant from focus and directrix.
3. The latus rectum of y² = 4ax has length:
Explanation: The latus rectum is the chord through the focus perpendicular to the axis. For y² = 4ax: at x = a, y² = 4a² → y = ±2a. Length = 4a.
4. The tangent to y² = 4ax at point (at², 2at) is:
Explanation: Tangent at parametric point (at², 2at) on y²=4ax: ty = x + at². Alternatively at point (x₁,y₁): yy₁ = 2a(x+x₁).
5. The normal to y² = 4ax at (at², 2at) has slope:
Explanation: Tangent at (at², 2at) has slope = 1/t (from differentiating y² = 4ax: dy/dx = 2a/y = 1/t). Normal slope = −1/(tangent slope) = −t.
6. From external point (h, k) to y² = 4ax, the chord of contact is:
Explanation: Chord of contact from (h,k) to y²=4ax: T = 0 → ky = 2a(x + h). This is the equation of the chord joining the two points of tangency.
7. The parabola y² = 4x and x² = 4y intersect at:
Explanation: Substitute y² = 4x into x² = 4y: x⁴/16 = 4(x/2) → x⁴ = 32x → x(x³ − 32) = 0 → x = 0 or x = 32^(1/3) ≈ 3.17. Wait: x² = 4y and y² = 4x. From first: y = x²/4. Sub in second: (x²/4)² = 4x → x⁴/16 = 4x → x³ = 64 → x = 4. So intersections at (0,0) and (4, 4).
8. All light rays parallel to the axis of y² = 4ax, after reflection:
Explanation: The reflective property of parabola: all rays parallel to the axis reflect off the parabola and pass through the focus. This is why parabolic reflectors (satellite dishes, car headlights) use this shape.
9. The foot of perpendicular from focus (a, 0) to the tangent ty = x + at² is:
Explanation: The foot of the perpendicular from the focus (a, 0) to any tangent ty = x + at² lies on the tangent at the vertex (the y-axis, x = 0). Substituting x = 0 in ty = x + at²: y = at. So foot = (0, at), which lies on x = 0 (tangent at vertex).
10. The vertex of y² = 8x is:
Explanation: Standard parabola y² = 4ax has vertex at the origin (0, 0). Here 4a = 8 → a = 2, focus at (2, 0).
11. The axis of symmetry of x² = 12y is:
Explanation: x² = 12y is symmetric about the y-axis (axis = y-axis). Vertex at origin, opens upward.
12. The length of the common chord of y² = 4x and x² = 4y is:
Explanation: The two parabolas intersect at (0,0) and (4,4). Length of common chord = distance between intersection points = √((4−0)²+(4−0)²) = √32 = 4√2.