Show, using algebra, that the formula for a parabola is a special case of the formula for an ellipse.

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Michael Ugulini | (Level 3) Educator

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The formula (standard equation) for an "ellipse" is:

     (x2/A2) + (y2/B2) = 1

Note that "a" is 1/2 the length of the major axis;"b" is 1/2 the length of the minor axis. The area of this ellipse is Pi * A * B.

Next, the expression of the equation of a "parabola" can be in vertex or standard form:

Vertex form is: y=a(x-h)² + k

Standard form is: y=ax² +bx + c

This all relates to conic sections whereby the conic section is the intersection of a plane and a cone. You can create a circle, ellipse, parabola or a hyperbola by changing the angle and location of intersection. Therfore, the above formulas address this altering via algebraic expression.

The ellipse formula as well as the parabola formula deal with the semi-major axis of length A and semi-minor axis of length B. The general conic section equation is:

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0

 

... and the type of section is located from the sign of B² - 4AC.

Therefore, if this B² - 4AC is less than 0 than the curve is an ellipse. If it equals 0 than it is a parabola. If it is greater than 0 than it is a hyperbola with two intersecting lines.

Therefore, with the above algebraic equations you can see the interrelation between ellipses and parabolas.

 

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