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

It can be shown that the value

$b$ from the previous question relates to the vertices of the minor-axis. Specifically, the vertices are

$(0,-b)$ and

$(0,b)$ . Looking at the positive vertex, it forms an isosceles triangle with the two foci. What is the length of the two congruent sides? How is this related to the previous question?

They each have a length of $2c$ . Using the distance formula and looking at the difference of distances between the lengths just found and the other vertices of the major axes, one finds that $b = sqrt(a^2-c^2$ .
They each have a length of $c$ . Then, the right triangle formed between the vertices and the origin, and applying the Pythagorean theorem, results in $a^2 + b^2 = c^2$ .
They each have a length of $a$ . Looking at the right triangle formed by the origin, $F_1$ , and the vertex $(0,b)$ , and applying the Pythagorean theorem results in $a^2 = b^2 + c^2$ .
They each have a length of $a/2$ . Therefore, the sum of their lengths, $a$ , can be used as a value equal to the sum of the lengths of $b$ and $c$ .

Question Info

Nonlinear Equations and Functions Question

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