CONIC SECTIONS
Frequently Asked Questions
A conic section is the curve obtained by the intersection of a plane with a right circular cone. Depending on the inclination of the plane, the curve may be a circle, parabola, ellipse, or hyperbola.
The curves included are circle, parabola, ellipse, and hyperbola.
A conic is the locus of a point such that the ratio of its distance from a fixed point (focus) to its distance from a fixed line (directrix) is constant.
Eccentricity \(e\) is the constant ratio of the distance of any point on the conic from the focus to its distance from the directrix.
If \(e=0\), the conic is a circle; if \(e=1\), a parabola; if \(0<e<1\), an ellipse; if \(e>1\), a hyperbola.
The standard equation is \(x^2+y^2=r^2\), where \(r\) is the radius.
The general equation is \(x^2+y^2+2gx+2fy+c=0\).
The center is \((-g,-f)\) and the radius is \(\sqrt{g^2+f^2-c}\), provided \(g^2+f^2-c>0\).
A circle is real if \(g^2+f^2-c>0\).
A parabola is the locus of a point whose distance from a fixed point equals its distance from a fixed line.
The standard equation is \(y^2=4ax\).
The focus is \((a,0)\).
The directrix is \(x=-a\).
The length of the latus rectum is \(4a\).
An ellipse is the locus of a point such that the sum of its distances from two fixed points is constant.