Conic Sections — Class 11 MCQs with Answers

Conic Sections — important questions & MCQs with answers (Class 11 Mathematics)

6 solved questions from this chapter's difficulty levels, each with its answer and explanation. Tap an option to check yourself.

0 of 6 answered

1. Q1 of 6Easy

   Equation of circle centre (0,0), radius r:

   Ax²+y²=r²Bx²−y²=r²Cx+y=rDx²+y² = r
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   Answer: x²+y²=r²

   Explanation: Every point stays a fixed distance r from the centre, and squaring that distance gives x²+y²=r² — squaring both sides is the step everyone forgets, which is why x²+y²=r (missing the square on r) tempts.

2. Q2 of 6Easy

   A circle is the set of all points:

   AEquidistant from a fixed pointBOn a straight lineCFrom a parabolaDOf finite distance from origin
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   Answer: Equidistant from a fixed point

   Explanation: A circle is the locus of points at a fixed distance (the radius) from one fixed point (the centre) — an ellipse instead keeps the SUM of distances to two foci constant, a definition students often blend with this one.

3. Q3 of 6Medium

   Focus of parabola y² = 12x:

   A(0, 3)B(3, 0)C(−3, 0)D(0, −3)
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   Answer: (3, 0)

   Explanation: 4a=12 gives a=3, and the focus of y²=4ax sits at (a,0)=(3,0) — mixing up focus and directrix would instead give x=−3.

4. Q4 of 6Medium

   Assertion (A): For the ellipse x²/16 + y²/25 = 1, the major axis lies along the x-axis. Reason (R): In the standard equation of an ellipse, the major axis is along the axis of the variable that has the larger denominator.

   ABoth A and R are true and R is the correct explanation of ABBoth A and R are true but R is not the correct explanation of ACA is true but R is falseDA is false but R is true
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   Answer: A is false but R is true

   Explanation: R is true (NCERT observation 2). Here the larger denominator, 25, is under y², so the major axis lies along the y-axis. Hence A is false.

5. Q5 of 6Hard

   For hyperbola x²/16 − y²/9 = 1, length of conjugate axis:

   A4B8C6D18
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   Answer: 6

   Explanation: b²=9 gives b=3, and the conjugate axis is 2b=6 — the transverse axis 2a=8 is a different, tempting number since a²=16.

6. Q6 of 6Hard

   Assertion (A): An equilateral triangle is inscribed in the parabola y² = 4ax with one vertex at the vertex of the parabola. The length of each side of the triangle is 8√3a. Reason (R): The other two vertices of this triangle are the end points of the latus rectum of the parabola.

   ABoth A and R are true and R is the correct explanation of ABBoth A and R are true but R is not the correct explanation of ACA is true but R is falseDA is false but R is true
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   Answer: A is true but R is false

   Explanation: By symmetry the two sides from the origin make 30° with the axis, so one side lies on y = x/√3. Putting this in y² = 4ax gives x²/3 = 4ax, so x = 12a and y = ±4√3a. Each side is 2 × 4√3a = 8√3a, so A is true. These vertices lie on x = 12a, but the latus rectum lies on x = a, so R is false.

Conic Sections notes: key concepts (Class 11 Mathematics)

Slicing a double cone gives the conic sections: circle, parabola, ellipse and hyperbola. This chapter derives each standard equation and defines eccentricity e as the number that classifies them — 0 for a circle, below 1 ellipse, 1 parabola, above 1 hyperbola — then locates the foci, directrices and latus rectum.

Conic sections

Curves formed when a plane cuts a double right circular cone; the tilt of the plane produces a circle, ellipse, parabola or hyperbola.

Circle

The locus of points at a fixed distance r (the radius) from a fixed centre (h, k); its equation is (x − h)² + (y − k)² = r².

Eccentricity

The ratio e measuring how far a conic departs from a circle: 0 for a circle, e < 1 ellipse, e = 1 parabola and e > 1 hyperbola.

Parabola

The locus of points equidistant from a fixed focus and a fixed directrix; the standard form y² = 4ax opens rightward with e = 1.

Parabola focus and directrix

For y² = 4ax the focus is (a, 0), the directrix is x = −a, the axis is the x-axis and the latus rectum has length 4a.

Four parabola orientations

y² = 4ax, y² = −4ax, x² = 4ay and x² = −4ay open right, left, up and down respectively, each with its vertex at the origin.

Ellipse

The locus where the sum of distances from two foci is constant; x²/a² + y²/b² = 1 with a > b, e < 1 and foci on the major axis.

Ellipse parameters

For x²/a² + y²/b² = 1 with a > b, the relation b² = a²(1 − e²) holds, foci are (±ae, 0) and the latus rectum is 2b²/a.

Hyperbola

The locus where the difference of distances from two foci is constant; x²/a² − y²/b² = 1 with e > 1 and foci beyond the vertices.

Hyperbola parameters

For x²/a² − y²/b² = 1 the relation b² = a²(e² − 1) holds, foci are (±ae, 0) and the latus rectum length is 2b²/a.

Latus rectum

The focal chord perpendicular to the conic's axis (the major axis of an ellipse, the transverse axis of a hyperbola); it measures 4a for y² = 4ax and 2b²/a for an ellipse or a hyperbola.

Degenerate conics

When the cutting plane passes through the cone's apex, the section collapses to a point, a single line or a pair of intersecting lines.