Conic Sections Questions with Solutions

11 questions3 easy6 medium2 hard

Conic sections are the curves formed when a plane cuts a double cone: the circle, parabola, ellipse and hyperbola. They can all be defined using a focus, a directrix and the eccentricity e, which is the ratio of a point's distance from the focus to its distance from the directrix. A conic is a parabola when e = 1, an ellipse when e < 1, and a hyperbola when e > 1. This topic brings the three conics together and tests identifying a conic from its equation, comparing their standard forms, and finding the focus, directrix, eccentricity and latus rectum. Mixed questions often ask you to recognise which conic an equation represents before applying the right formulas. For in-depth practice on each curve, see the separate parabola, ellipse and hyperbola pages. Practise the conic section questions below, each with a step-by-step solution.

All Questions Easy Medium Hard

Q1.
A circle touches the x-axis and also touches the circle with centre $(0, 3)$ and radius 2. The locus of the centre of the circle is
A. a circleB. an ellipseC. a parabolaD. a hyperbola
mediumAsked in NIMCET 2023View solution ›
Q2.
The focus of the parabola $y^{2} = 12 x$ is
A. (3, 0)B. (0, 3)C. (12, 0)D. (−3, 0)
easyView solution ›
Q3.
The eccentricity of the ellipse $\frac{x ^{2}}{25} + \frac{y ^{2}}{16} = 1$ is
A. $\frac{4}{5}$ B. $\frac{3}{5}$ C. $\frac{3}{4}$ D. $\frac{5}{3}$
easyView solution ›
Q4.
The length of the latus rectum of the parabola $x^{2} = - 16 y$ is
A. 4B. 8C. 16D. −16
easyView solution ›
Q5.
The centre and radius of the circle $x^{2} + y^{2} - 4 x + 6 y - 12 = 0$ are
A. (2, −3) and 5B. (−2, 3) and 5C. (2, −3) and 25D. (4, −6) and 5
mediumView solution ›
Q6.
Find the number of point(s) of intersection of the ellipse $\frac{x ^{2}}{4} + \frac{( y - 1 ) ^{2}}{9} = 1$ and the circle $x^{2} + y^{2} = 4$
A. 4B. 3C. 2D. 1
mediumAsked in NIMCET 2020View solution ›
Q7.
If the foci of the ellipse $\frac{x ^{2}}{25} + \frac{y ^{2}}{b ^{2}} = 1$ and the hyperbola $\frac{x ^{2}}{144} - \frac{y ^{2}}{81} = \frac{1}{25}$ coincide, then the value of $b^{2}$ is
A. 25B. 16C. 64D. 49
mediumAsked in NIMCET 2022View solution ›
Q8.
If $(\frac{x}{a})^{2} + (\frac{y}{b})^{2} = 1$ $(a > b)$ and $x^{2} - y^{2} = c^{2}$ cut at right angles, then
A. $a^{2} + b^{2} = 2 c^{2}$ B. $b^{2} - a^{2} = 2 c^{2}$ C. $a^{2} - b^{2} = 2 c^{2}$ D. $a^{2} - b^{2} = c^{2}$
hardAsked in NIMCET 2022View solution ›
Q9.
If the foci of the ellipse $b^{2} x^{2} + 16 y^{2} = 16 b^{2}$ and the hyperbola $81 x^{2} - 144 y^{2} = \frac{81 \times 144}{25}$ coincide, then the value of b, is
A. $1$ B. $5$ C. $7$ D. $3$
mediumAsked in NIMCET 2014View solution ›
Q10.
At how many points do the curves $\frac{y ^{2}}{9} - \frac{x ^{2}}{16} = 1$ and $\frac{x ^{2}}{4} + \frac{( y - 4 ) ^{2}}{16} = 1$ intersect?
A. 0B. 1C. 2D. 4
hardAsked in NIMCET 2024View solution ›
Q11.
The equation $3 x^{2} + 10 x y + 11 y^{2} + 14 x + 12 y + 5 = 0$ represents
A. a circleB. an ellipseC. a hyperbolaD. a parabola
mediumAsked in NIMCET 2024View solution ›