Circles and Parabolas

Front

What is the definition of a circle in a coordinate plane?

Back

A circle is the set of all points in a plane that are equidistant from a fixed point called the center. The standard equation of a circle with center (h, k) and radius r is (x - h)² + (y - k)² = r².

Front

What is the standard form of a parabola that opens upwards?

Back

The standard form of a parabola that opens upwards is given by the equation y = a(x - h)² + k, where (h, k) is the vertex of the parabola.

Front

What is the focus of a parabola?

Back

The focus of a parabola is a fixed point located inside the curve, which is used to define the parabola. It is equidistant from the directrix and any point on the parabola.

Front

What is the directrix of a parabola?

Back

The directrix of a parabola is a fixed line used in the definition of the parabola. It is located opposite the focus and is perpendicular to the axis of symmetry.

Front

How do you find the radius of a circle from its equation?

Back

To find the radius of a circle from its equation in standard form (x - h)² + (y - k)² = r², take the square root of r².

Front

What is the relationship between the focus and directrix of a parabola?

Back

For any point on the parabola, the distance to the focus is equal to the distance to the directrix.

Front

What is the equation of a circle with center at (0, 0) and radius 3?

Back

The equation of the circle is x² + y² = 9.

Front

What is the vertex of a parabola?

Back

The vertex of a parabola is the highest or lowest point on the graph, depending on the direction it opens. It is represented by the point (h, k) in the standard form of the parabola.

Front

What does the term 'conic sections' refer to?

Back

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. The main types are circles, ellipses, parabolas, and hyperbolas.

Front

What is the significance of the parameter 'a' in the parabola equation y = a(x - h)² + k?

Back

The parameter 'a' determines the direction and width of the parabola. If 'a' is positive, the parabola opens upwards; if negative, it opens downwards. The larger the absolute value of 'a', the narrower the parabola.