Question

a) in the illustration we have placed the focal points on the x - axis equidistant from the origin. the distance from the origin to the edge of the ellipse on the x - axis is labeled a in the figure. this is the length of the semi - major axis. find the sum of the distances from the focal points to the point (a,0).
b) derive a relation between a, b (the length of the semi - minor axis), and c by finding the sum of the distances from the focal points to the point (0,b) and setting it equal to your result from part a).
c) using the distance formula in eq. (11.21) write down the sum of the distances from the focal points to a generic point (x,y) on the ellipse. use your relation from part b) to show that the equation for the ellipse is: $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1$ (12.30)
in the case where a = b the ellipse becomes a circle of radius a.
d) the eccentricity of an ellipse is defined as e = c/a < 1. if you designated the point on the ellipse above the focal point at (c,0) to be at (c,y_c), show that the following relations hold $a=\frac{y_c}{1 - e^{2}}$ and $b=\frac{y_c}{sqrt{1 - e^{2}}}$ (12.31)
what is the eccentricity for a circle?
e) by completing squares show that the following

Explanation:

Step1: Recall ellipse property

For an ellipse with foci $F_1(-c,0)$ and $F_2(c,0)$, the sum of the distances from the foci to any point on the ellipse is a constant. For the point $(a,0)$, the distance from $F_1(-c,0)$ to $(a,0)$ is $a + c$ and from $F_2(c,0)$ to $(a,0)$ is $a - c$. The sum is $(a + c)+(a - c)=2a$.

Step2: Consider point (0,b)

The distance from $F_1(-c,0)$ to $(0,b)$ is $\sqrt{(0 + c)^2+(b - 0)^2}=\sqrt{c^{2}+b^{2}}$, and from $F_2(c,0)$ to $(0,b)$ is also $\sqrt{(0 - c)^2+(b - 0)^2}=\sqrt{c^{2}+b^{2}}$. The sum is $2\sqrt{c^{2}+b^{2}}$. Setting it equal to the result from part a) ($2a$), we get $2\sqrt{c^{2}+b^{2}}=2a$, so $a^{2}=b^{2}+c^{2}$.

Step3: For general point (x,y)

The distance from $F_1(-c,0)$ to $(x,y)$ is $d_1=\sqrt{(x + c)^2+y^{2}}$ and from $F_2(c,0)$ to $(x,y)$ is $d_2=\sqrt{(x - c)^2+y^{2}}$. The sum $d_1 + d_2$:
\[

$$\begin{align*} d_1 + d_2&=\sqrt{(x + c)^2+y^{2}}+\sqrt{(x - c)^2+y^{2}}\\ \end{align*}$$

\]
Using $a^{2}=b^{2}+c^{2}$, we start with the definition of an ellipse $d_1 + d_2 = 2a$. Square both sides:
\[

$$\begin{align*} (\sqrt{(x + c)^2+y^{2}}+\sqrt{(x - c)^2+y^{2}})^2&=(2a)^2\\ (x + c)^2+y^{2}+2\sqrt{[(x + c)^2+y^{2}][(x - c)^2+y^{2}]}+(x - c)^2+y^{2}&=4a^{2}\\ x^{2}+2cx + c^{2}+y^{2}+2\sqrt{(x^{2}+2cx + c^{2}+y^{2})(x^{2}-2cx + c^{2}+y^{2})}+x^{2}-2cx + c^{2}+y^{2}&=4a^{2}\\ 2(x^{2}+y^{2}+c^{2})+2\sqrt{(x^{2}+y^{2}+c^{2})^{2}-4c^{2}x^{2}}&=4a^{2}\\ \sqrt{(x^{2}+y^{2}+c^{2})^{2}-4c^{2}x^{2}}&=2a^{2}-(x^{2}+y^{2}+c^{2}) \end{align*}$$

\]
Square both sides again and simplify using $a^{2}-c^{2}=b^{2}$ to get $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1$.

Step4: For point (c,y_c) on ellipse

Since $(c,y_c)$ lies on the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1$, substituting $x = c$ gives $\frac{c^{2}}{a^{2}}+\frac{y_{c}^{2}}{b^{2}} = 1$. Using $e=\frac{c}{a}$, so $c = ae$ and $1 - e^{2}=\frac{a^{2}-c^{2}}{a^{2}}=\frac{b^{2}}{a^{2}}$.
\[

$$\begin{align*} \frac{e^{2}a^{2}}{a^{2}}+\frac{y_{c}^{2}}{b^{2}}&=1\\ \frac{y_{c}^{2}}{b^{2}}&=1 - e^{2}\\ y_{c}^{2}&=b^{2}(1 - e^{2}) \end{align*}$$

\]
Also, from $a^{2}=b^{2}+c^{2}$ and $c = ae$, we can show $a=\frac{y_{c}}{1 - e^{2}}$ and $b=\frac{y_{c}}{\sqrt{1 - e^{2}}}$. For a circle, $a = b$ and $c = 0$, so the eccentricity $e=\frac{c}{a}=0$.

Step5: Completing - squares (not shown in full here as the problem doesn't fully state what to complete - square for, but the general idea for an ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1$ is not relevant for the previous derivations)

Answer:

a) The sum of the distances from the focal points to the point $(a,0)$ is $2a$.
b) The relation is $a^{2}=b^{2}+c^{2}$.
c) The derivation shows the standard equation of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1$.
d) For a circle, the eccentricity $e = 0$.
e) (No full - answer provided as the problem statement for part e is incomplete in the reference to what needs to be completed - squared).