Question

Please Help!!!
Due: Monday
On Ellipses - Pre Calc

Answer 1

25. Explanation:

`\text{The general form of an ellipse is:}((x-h)^2)/(a^2)+((y-k)^2)/(b^2)=1\\\\\bullet \text{(h, k) is the Center}\\\bullet \text{a is the radius of x}\\\bullet \text{b is the radius of y}\\\bullet \text{the largest value between a and b is the major}\\\bullet \text{the smallest value between a and b is the minor}\\\bullet \text{the vertices are the (h, k) value plus the major (a or b) value}\\\bullet \text{the co-vertices are the (h, k) value plus the minor (a or b) value}\\\bullet \text{Length is the diameter}=2r\\`

`(x^2)/(16)+(y^2)/(25)=1\quad \text{can be rewritten as}\ ((x-0)^2)/(4^2)+((y-0)^2)/(5^2)=1\\\\\bullet (h, k)=(0,0)\\\bullet a=4\\\bullet b=5\\\bullet \text{b is the largest value so: b is the major and a is the minor}\\\bullet \text{Vertices are }(0, 0+5)\ and\ (0, 0-5)\implies (0, 5)\ and\ (0, -5)\\\bullet \text{Co-vertices are }(0+4, 0)\ and\ (0-4, 0)\implies (4,0)\ and\ (-4,0)\\\bullet \text{Length of major is }2b:2(5)=10\\\bullet \text{Length of minor is }2a:2(4)=8`

To find the foci, first we must find the length of the foci using the formula:

`(r_(major))^2-(r_(minor))^2=c^2`

Then add the c-value to the h (or k)-value that represents the major.

b² - a² = c²

25 - 16 = c²

9 = c²

±3 = c

The center is (0, 0) and the major is the y-value so the foci is:

(0, 0+3) and (0, 0-3) ⇒ (0, 3) and (0, -3)

26. Answers

Follow the same steps as #25:

Center: (0, 0)

Vertices (7, 0) and (-7, 0)

Co-vertices: (0, 3) and (0, -3)

foci: (2√10, 0) and (-2√10, 0)

length of major: 14

length of minor: 6