68 free MCQs on Conic Sections, each with its own worked answer and explanation. Study circles, parabolas, ellipses, and hyperbolas as curves formed by intersecting a plane with a double cone, with their standard equations and key properties.
Take the timed Conic Sections chapterwise test →68 practice questions on Conic Sections, sorted Easy → Hard. Try each one first, then open its answer page for the worked explanation. Want the full theory first? Read the Conic Sections notes.
All four conics form a single family distinguished only by eccentricity: a circle is the most "closed" (e=0), an ellipse is an elongated closed curve, a parabola is the borderline open curve (e=1), and a hyperbola has two separate open branches (e>1).
Q1.
A conic section is formed by the intersection of a plane with:
Q2.
The eccentricity of a circle is:
Q3.
The eccentricity of a parabola is always:
Q4.
For an ellipse, the eccentricity e satisfies:
Q5.
For a hyperbola, the eccentricity e satisfies:
Q6.
The standard equation of a parabola opening to the right with vertex at the origin is:
Q7.
For the parabola y<sup>2</sup> = 4ax, the coordinates of the focus are:
Q8.
The standard equation of an ellipse with major axis along the x-axis is:
Q9.
The standard equation of a hyperbola with transverse axis along the x-axis is:
Q10.
For an ellipse x<sup>2</sup>/a<sup>2</sup> + y<sup>2</sup>/b<sup>2</sup> = 1 with a > b, the relationship between a, b, and c (distance to focus) is:
Q11.
For a hyperbola x<sup>2</sup>/a<sup>2</sup> - y<sup>2</sup>/b<sup>2</sup> = 1, the relationship between a, b, and c (distance to focus) is:
Q12.
The length of the latus rectum of the parabola y<sup>2</sup> = 4ax is:
Q13.
The general equation of a circle with center (h, k) and radius r is:
Q14.
A circle, ellipse, parabola and hyperbola are together known as:
Q15.
The standard equation of a circle with centre at the origin and radius r is:
Q16.
The fixed point used to define a parabola is called its:
Q17.
The standard equation of a parabola opening to the right is:
Q19.
The longest diameter of an ellipse is called the:
Q20.
A hyperbola consists of two branches and has how many foci?
Q21.
Find the focus of the parabola y<sup>2</sup> = 12x.
Q22.
Find the length of the latus rectum of the parabola y<sup>2</sup> = 12x.
Q23.
Find the equation of the directrix of the parabola y<sup>2</sup> = 12x.
Q24.
For the ellipse x<sup>2</sup>/25 + y<sup>2</sup>/9 = 1, find the eccentricity.
Q25.
For the ellipse x<sup>2</sup>/25 + y<sup>2</sup>/9 = 1, find the coordinates of the foci.
Q26.
For the ellipse x<sup>2</sup>/25 + y<sup>2</sup>/9 = 1, find the length of the latus rectum.
Q27.
For the hyperbola x<sup>2</sup>/16 - y<sup>2</sup>/9 = 1, find the eccentricity.
Q28.
For the hyperbola x<sup>2</sup>/16 - y<sup>2</sup>/9 = 1, find the equations of the asymptotes.
Q29.
Find the equation of a parabola with vertex at the origin and focus at (5, 0).
Q30.
An ellipse has vertices (±5, 0) and foci (±3, 0). Find its equation.
Q31.
A hyperbola has vertices (±4, 0) and foci (±6, 0). Find the value of b<sup>2.</sup>
Q32.
Find the directrices of the ellipse x<sup>2</sup>/25 + y<sup>2</sup>/9 = 1.
Q33.
Identify the conic represented by 9x<sup>2</sup> + 25y<sup>2</sup> = 225 and find its eccentricity.
Q34.
For the ellipse x²/a² + y²/b² = 1 with a > b, the eccentricity e is:
Q35.
For a parabola, the eccentricity e equals:
Q36.
For a hyperbola, the eccentricity e is:
Q37.
The equation x²/16 + y²/9 = 1 represents an:
Q38.
The equation x²/16 − y²/9 = 1 represents a:
Q39.
For the parabola y² = 4ax, the focus is located at:
Q40.
The length of the latus rectum of the parabola y² = 4ax is:
Q41.
Find the equation of the parabola with vertex at the origin, axis along the x-axis, and passing through the point (2, 4).
Q42.
An ellipse has eccentricity 3/5 and its foci at (±3, 0). Find the equation of the ellipse.
Q43.
Find the length of the latus rectum and the eccentricity of the hyperbola 9x<sup>2</sup> - 16y<sup>2</sup> = 144.
Q44.
A point on a parabola y<sup>2</sup> = 8x is at a distance of 6 units from the focus. Find its distance from the directrix.
Q45.
Find the equation of an ellipse whose major axis is along the y-axis, with semi-major axis 5 and semi-minor axis 3.
Q46.
Find the distance between the directrices of the hyperbola x<sup>2</sup>/9 - y<sup>2</sup>/16 = 1.
Q47.
The latus rectum of an ellipse is half of its minor axis. Find the eccentricity of the ellipse.
Q48.
Find the eccentricity of the hyperbola whose latus rectum is equal to half of its transverse axis.
Q49.
A line passes through the focus of the parabola y<sup>2</sup> = 4ax and is perpendicular to its axis. The chord it cuts on the parabola is called the latus rectum. If a = 4, find the endpoints of the latus rectum.
Q50.
Find the equation of the hyperbola with foci (±5, 0) and transverse axis of length 8.
Q51.
An ellipse and a hyperbola have the same foci. If the ellipse has eccentricity 3/5 and the hyperbola has eccentricity 5/3, and the ellipse's semi-major axis is 10, find c (distance from center to focus).
Q52.
The eccentricity of a conic is found to be exactly 1. Which type of conic must it be, and what defines its shape uniquely?
Q53.
For the ellipse x²/25 + y²/16 = 1, the value of a (semi-major axis) is:
Q54.
For the ellipse x²/25 + y²/16 = 1, the eccentricity e = √(1 − b²/a²) equals:
Q55.
The directrix of the parabola y² = 4ax is the line:
Q56.
A general second-degree equation represents a circle when the coefficients of x² and y² are equal and the xy term is:
Q57.
The vertices of the hyperbola x²/a² − y²/b² = 1 are located at:
Q58.
The centre of the circle x² + y² − 6x + 4y − 12 = 0 is:
Q60.
For an ellipse, the sum of the distances from any point on it to the two foci equals:
Q61.
The eccentricity of the ellipse x²/25 + y²/16 = 1 is:
Q62.
The length of the latus rectum of the parabola y² = 12x is:
Q63.
The foci of the hyperbola x²/9 − y²/16 = 1 are:
Q64.
The directrix of the parabola x² = 8y is:
Q65.
The eccentricity of a rectangular hyperbola (asymptotes y = ±x) is:
Q66.
The line y = x + c is tangent to the circle x² + y² = 2 when c equals:
Q67.
The centre of the hyperbola (x − 1)²/4 − (y − 2)²/9 = 1 is:
Q68.
The sum of the focal distances of any point on the ellipse x²/16 + y²/9 = 1 is:
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