Exercise 34 Page 642

We want to write an equation of an ellipse with foci (0,± 4) and co-vertices (± 2,0). Note that the foci are on the y-axis, and therefore the major axis is vertical. This means the ellipse is a vertical ellipse. Let's recall the equation of a vertical ellipse with vertices (0,± a) and co-vertices (± b,0). x^2/b^2+y^2/a^2=1, a>b>0If we let the vertices of our ellipse be (0,± a), we can write and solve an equation to find the value of a.

c^2= a^2- b^2

SubstituteII

b= 2, c= 4

4^2= a^2- 2^2

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Solve for a

CalcPow

Calculate power

16=a^2-4

AddEqn

LHS+4=RHS+4

20=a^2

RadicalEqn

sqrt(LHS)=sqrt(RHS)

sqrt(20)=a

Rewrite

Rewrite 20 as 4 * 5

sqrt(4 * 5)=a

RootProd

sqrt(a* b)=sqrt(a)*sqrt(b)

sqrt(4) * sqrt(5)=a

CalcRoot

Calculate root

2sqrt(5)=a

RearrangeEqn

Rearrange equation

a= 2sqrt(5)

Note that we took the principal root, because a is a positive number. Now that we know the values of a and b, we can write the equation of our ellipse. x^2/2^2+y^2/( 2sqrt(5))^2=1 ⇔ x^2/4+y^2/20=1