Pre-Calculus Reviewer – Conic Sections & Circles

Question-and-answer flashcards that review definitions, classifications, equations, and problem-solving methods for conic sections and circles.

What are conic sections?

Curves produced when a plane intersects a double right circular cone.

What are the four primary conic sections?

Ellipse, circle, parabola, and hyperbola.

Which plane orientation produces a circle from a cone?

A plane perpendicular to the cone’s axis intersects one nappe.

Which plane orientation produces an ellipse?

A plane that intersects one nappe at an oblique (angled) angle.

Which plane orientation produces a parabola?

A plane parallel to a generator (slanted edge) of the cone.

Which plane orientation produces a hyperbola?

A plane that intersects both nappes of the cone.

When does a conic section degenerate into a single point?

When the cutting plane passes through the vertex of the cone.

When does a conic section degenerate into a single line?

When the plane touches one nappe at a slant.

When does a conic section degenerate into two intersecting lines?

When the plane vertically cuts through the vertex.

What is the axis of a double cone?

The vertical line of symmetry that runs through both nappes.

What is the generator of a cone?

The slanted edge that traces the cone’s surface as it revolves.

What is the directrix of a cone?

The perimeter (circumference) of the cone’s base.

What is the vertex angle of a cone?

The angle between the axis and a generator.

Define a circle in the plane.

The set of all points equidistant from a fixed point called the center.

What is the standard form equation of a circle centered at the origin?

x² + y² = r²

What is the standard form equation of a circle centered at (h, k)?

(x − h)² + (y − k)² = r²

What is the general form of a circle’s equation?

x² + y² + D x + E y + F = 0

How do you find a circle’s center from its general form?

Center (h, k) where h = −D⁄2 and k = −E⁄2.

How do you find a circle’s radius from its general form?

r = √(h² + k² − F) after computing h and k.

What formula gives the radius when you know the center and a point on the circle?

Distance formula: r = √[(x₂ − x₁)² + (y₂ − y₁)²].

If the endpoints of a diameter are given, how do you find the center and radius?

Center = midpoint of the endpoints; radius = half the distance between the endpoints.

If a circle is tangent to the x-axis or y-axis, how do you find the radius?

Radius equals the perpendicular (vertical or horizontal) distance from the center to that axis.

If a circle is tangent to the line y = c or x = c, how do you find the radius?

Radius equals the absolute perpendicular distance from the center to the given line using the point-to-line distance formula.