Nonlinear Equations and Functions Question

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For the ellipse with foci

$F_1 = (-c,0)$ and

$F_2 = (c,0)$ , derive the equation of the ellipse.

Grade 11 Nonlinear Equations and Functions CCSS: HSG-GPE.A.3

After some more algebra, and then moving all terms with

$x$ or

$y$ to the left side of the equation (and all others to the right), the resulting equation is:

$a^2 x^2 - c^2 x^2 + a^2 y^2$	$=$	$a^4 -a^2 c^2$
$(a^2 - c^2)x^2 + a^2 y^2$	$=$	$(a^2 - c^2) a^2$
$b^2 x^2 + a^2 y^2$	$=$	$a^2 b^2$
$x^2 / a^2 + y^2 / b^2$	$=$	$1$

This is the formula for the equation of an ellipse centered at

$(0,0)$ . Why can the substitution

$b^2 = a^2-c^2$ be made?

Since $b$ is undefined so far, it can be defined as any value. Then, using the Pythagorean theorem, $a^2 + b^2 = c^2$ , simply rearrange to solve for $b^2$ .
Knowing that the semi-minor axis is $b$ units long, one can substitute the square of this value for $a^2-c^2$ .
Since $b$ is not yet defined, it can be used to simplify the equation by defining $b^2 = a^2 - c^2$ . A positive value for $a^2-c^2$ exists since $|a| > |c|$ .
Projecting $b$ , the length of the semi-minor axis, onto the semi-major axis it is seen that $b=a-c$ . Then, simply square both sides of the equation.

Question Info

Nonlinear Equations and Functions Question

Grade 11 Nonlinear Equations and Functions CCSS: HSG-GPE.A.3