AP Precalc Conic Sections

What is a conic section?

A conic section is a curve obtained by intersecting a cone with a plane. The main types are circles, ellipses, parabolas, and hyperbolas.

What is the standard form equation of a circle?

The standard form is (x-h)^2 + (y-k)^2 = r^2, where (h, k) is the center and r is the radius.

What is the standard form equation of an ellipse?

The standard form is ((x-h)^2 / a^2) + ((y-k)^2 / b^2) = 1, where (h, k) is the center, a is the semi-major axis, and b is the semi-minor axis.

What is the standard form equation of a parabola?

The standard form is y = a(x-h)^2 + k or x = a(y-k)^2 + h, where (h, k) is the vertex.

What is the axis of symmetry of a parabola?

A line passing through the focus and perpendicular to the directrix of a parabola.

The distance from the center to a

Practice Problem: Find the equation of a circle with center (2, -3) and radius 4.

Using the standard form (x-h)^2 + (y-k)^2 = r^2, the equation is (x-2)^2 + (y+3)^2 = 16.

Practice Problem: Find the equation of an ellipse with center (0, 0), semi-major axis a = 5, and semi-minor axis b = 3.

Using the standard form ((x-h)^2 / a^2) + ((y-k)^2 / b^2) = 1, the equation is (x^2 / 25) + (y^2 / 9) = 1.

Practice Problem: Find the equation of a parabola with vertex (1, 2) and a = 2 that opens upwards.

Using the standard form y = a(x-h)^2 + k, the equation is y = 2(x-1)^2 + 2.

Practice Problem: Find the equation of a hyperbola with center (0, 0), a = 4, b = 3, opening along the x-axis.

Using the standard form ((x-h)^2 / a^2) - ((y-k)^2 / b^2) = 1, the equation is (x^2 / 16) - (y^2 / 9) = 1.