SS3 First Term Further Mathematics Lesson Note – Conic Sections

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WEEK 5        

TOPIC:                      Conic Sections

CONTENT:  

• Equations of parabola, Eclipse, Hyperbola in rectangular Cartesian coordinates
• Parametric equations

Sub-Topic (i): Equations of parabola, ellipse, hyperbola in rectangular coordinates

The path (or locus) of a point which moves so that its distance from a fixed point is in constant ratio to its distance from a fixed line is called a conic section or a conic.

The fixed point F is called the focus, the fixed line is called the directrix, and the constant ratio is called eccentricity, usually denoted by e.

Thus, if F is a point on the locus (as shown in the diagram above), M is the foot of the perpendicular from P to the directrix and if  , then the locus of F is a conic. If e<1, the conic is an ellipse. If e = 1, the conic is a parabola. If e>1, the conic is a hyperbola.

Parabola: a parabola is the locus of points in a plane which is equidistant from a fixed line and a fixed point. Let L be the fixed line and F the fixed point as shown below. Through F draw the x-axis perpendicular to the fixed line of distance 20 units from F. From the definition of the parabola, the curve must cross the x-axis at a point 0, midway between F and L. Then draw the y-axis through 0. The co-ordinates of F are (a, 0).

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