Parabola Questions with Solutions

13 questions2 easy9 medium2 hard

A parabola is the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix), so its eccentricity is exactly 1. The standard form y^2 = 4ax has its vertex at the origin, focus at (a, 0), directrix x = -a, and latus rectum of length 4a. The other standard forms open left, up or down. Exam questions test identifying the focus, directrix, axis and latus rectum, the parametric point (at^2, 2at), focal distances, and equations of tangents and normals. A key result is that the line y = mx + c touches y^2 = 4ax when c = a/m. Drawing a quick sketch of which way the parabola opens prevents most sign errors. Practise the parabola questions below, each with a step-by-step solution.

All Questions Easy Medium Hard

Q1.
Which of the following equations can represent a common tangent to the parabolas $y = - x^{2}$ and $y = (x - 2)^{2}$ ?
A. y = 4x + 4B. $y = 5 x + \frac{25}{4}$ C. y = −4x + 4D. $y = - 5 x + \frac{25}{4}$
mediumAsked in NIMCET 2026View solution ›
Q2.
The equation of a parabola is $y^{2} = 4 x$ . Which of the following points lies on the parabola?
A. $(4, 4)$ and $(4, - 4)$ B. $(3, 5)$ and $(3, - 5)$ C. $(1, 3)$ and $(1, - 3)$ D. $(6, 8)$ and $(6, - 8)$
mediumAsked in MAH MCA CET 2025View solution ›
Q3.
The directrix of the parabola $4 y^{2} + 12 x - 12 y + 39 = 0$ is
A. $x = \frac{3}{4}$ B. $x = - \frac{7}{4}$ C. $x = - \frac{5}{2}$ D. $x = \frac{3}{2}$
mediumAsked in AMU MCA 2016View solution ›
Q4.
A point P in the first quadrant lies on $y^{2} = 4 a x$ , $a > 0$ , and is at a distance $5 a$ from its focus. Which of the following points lies on the locus of P?
A. (1, 0)B. (1, 1)C. (0, 2)D. (2, 0)
mediumAsked in NIMCET 2023View solution ›
Q5.
The coordinates of the focus of the parabola $4 y^{2} + 12 x - 20 y + 67 = 0$ are
A. $(- \frac{5}{4}, \frac{17}{2})$ B. $(- \frac{17}{2}, \frac{5}{4})$ C. $(- \frac{17}{4}, \frac{5}{2})$ D. $(- \frac{5}{2}, \frac{17}{4})$
mediumAsked in NIMCET 2022View solution ›
Q6.
An equilateral triangle is inscribed in the parabola $y^{2} = 4 a x$ , such that one of the vertices of the triangle coincides with the vertex of the parabola. The length of the side of the triangle is:
A. $a 3$ B. $2 a 3$ C. $4 a 3$ D. $8 a 3$
mediumAsked in NIMCET 2014View solution ›
Q7.
If $x = 1$ is the directrix of the parabola $y^{2} = k x - 8$ , then k is:
A. $\frac{1}{8}$ B. $8$ C. $4$ D. $\frac{1}{4}$
mediumAsked in NIMCET 2014View solution ›
Q8.
A normal to the curve $x^{2} = 4 y$ passes through the point (1, 2). The distance of the origin from the normal is
A. $2$ B. $22$ C. $\frac{1}{2}$ D. $\frac{3}{2}$
mediumAsked in NIMCET 2014View solution ›
Q9.
An equilateral triangle is inscribed in the parabola $y^{2} = x$ . One vertex of the triangle is at the vertex of the parabola. The centroid of the triangle is
A. $(1, 0)$ B. $(2, 0)$ C. $(3, 0)$ D. $(2, 0)$
mediumAsked in NIMCET 2025View solution ›