Conics

Front

What is a conic section?

Back

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

Front

What is the general equation of a circle?

Back

The general equation of a circle is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.

Front

What is the standard form of an ellipse?

Back

The standard form of an ellipse is (x - h)²/a² + (y - k)²/b² = 1, where (h, k) is the center, a is the semi-major axis, and b is the semi-minor axis.

Front

What is the standard form of a hyperbola?

Back

The standard form of a hyperbola is (x - h)²/a² - (y - k)²/b² = 1 or (y - k)²/b² - (x - h)²/a² = 1, where (h, k) is the center.

Front

What is the definition of a parabola?

Back

A parabola is a conic section that is formed by the intersection of a cone and a plane parallel to its side. It can be represented by the equation y = ax² + bx + c.

Front

How do you find the vertex of a parabola given its equation in standard form?

Back

For the equation y = ax² + bx + c, the vertex can be found using the formula x = -b/(2a) and then substituting x back into the equation to find y.

Front

What are the foci of an ellipse?

Back

The foci of an ellipse are two fixed points located along the major axis, and the sum of the distances from any point on the ellipse to the foci is constant.

Front

What is the relationship between the eccentricity and the type of conic section?

Back

Eccentricity (e) determines the type of conic: e = 0 for a circle, 0 < e < 1 for an ellipse, e = 1 for a parabola, and e > 1 for a hyperbola.

Front

What is the equation of a hyperbola centered at the origin?

Back

The equation of a hyperbola centered at the origin is x²/a² - y²/b² = 1 or y²/b² - x²/a² = 1.

Front

How do you determine if a conic section is a circle, ellipse, or hyperbola from its equation?

Back

By analyzing the coefficients of x² and y²: If they are equal and have the same sign, it's a circle; if they are equal but have different signs, it's a hyperbola; if they are different but have the same sign, it's an ellipse.