Mathematics · Quantitative Aptitude

Conic Sections

245 Questions

Conic sections deal with the curves obtained by the intersection of a cone with a plane, primarily focusing on parabolas, ellipses, and hyperbolas. Questions require finding vertices, directrices, and asymptotes based on given equations. This is an advanced geometry topic for rigorous competitive exams.

Parabola equationsHyperbola propertiesEllipse conceptsTangents and normalsDirectrix and focus

Conic Sections Questions

Multiple choice

What is the standard form of the equation of a hyperbola?

  1. \(x^2 + y^2 = r^2\)
  2. \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)
  3. \(y^2 = 4px\)
  4. \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\)
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The standard form of the equation of a hyperbola is (\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1), where (a) and (b) are the lengths of the transverse and conjugate axes, respectively.

Multiple choice

What is the equation of the directrix of a parabola with vertex at the origin and focus at ((p, 0))?

  1. \(x = -p\)
  2. \(x = p\)
  3. \(y = -p\)
  4. \(y = p\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The equation of the directrix of a parabola with vertex at the origin and focus at ((p, 0)) is (x = -p).

Multiple choice

What is the equation of the directrix of a hyperbola with center at the origin and transverse axis (2a)?

  1. \(x = -2a\)
  2. \(x = 2a\)
  3. \(y = -2a\)
  4. \(y = 2a\)
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The equation of the directrix of a hyperbola with center at the origin and transverse axis (2a) is (x = 2a).

Multiple choice

Find the equation of the parabola with vertex at (0, 0) and focus at (0, 2).

  1. y^2 = 8x

  2. y^2 = 4x

  3. y^2 = 2x

  4. y^2 = x
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The standard equation of a parabola with vertex at the origin and focus at (0, p) is given by y^2 = 4px. Substituting p = 2, we get the equation y^2 = 4x.

Multiple choice

What is the equation of the hyperbola with center at the origin, vertices at (±3, 0), and foci at (±5, 0)?

  1. x^2 - y^2 = 9

  2. x^2 - y^2 = 25

  3. x^2 + y^2 = 9

  4. x^2 + y^2 = 25
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The standard equation of a hyperbola with center at the origin, vertices at (±a, 0), and foci at (±c, 0) is given by x^2 / a^2 - y^2 / b^2 = 1, where c^2 = a^2 + b^2. Substituting a = 3 and c = 5, we get: 5^2 = 3^2 + b^2, which gives b^2 = 16. Therefore, the equation of the hyperbola is x^2 / 9 - y^2 / 16 = 1, or equivalently, x^2 - y^2 = 25.