# Thread: [SOLVED] Length of the major axis?

1. ## [SOLVED] Length of the major axis?

heyy, (Barron SAT Math II Page 19)
what is the length of the major axis of the ellipse whose equation is 10x^2+20y^2=200?

a) 3.16
b)4.47
c) 6.32
d) 8.94
e) 14.14

The correct answer is D). The answer book said to divide both sides of the equation by 200 to write the equation in standard form. So then the length of the major axis is 2 $sqrt20$. Can please explain why? What does it mean by 'major' axis ?

2. Originally Posted by fabxx
heyy, (Barron SAT Math II Page 19)
what is the length of the major axis of the ellipse whose equation is 10x^2+20y^2=200?

a) 3.16
b)4.47
c) 6.32
d) 8.94
e) 14.14

The correct answer is D). The answer book said to divide both sides of the equation by 200 to write the equation in standard form. So then the length of the major axis is 2 $sqrt20$. Can please explain why? What does it mean by 'major' axis ?

An ellipse with it's center at the origin and the semiaxes a (=major semiaxis) and b(= minor semiaxis) has the equation:

$\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2}=1$

With the given equation you can easily calculate a. This value has to be doubled to get the complete major axis:

$10x^2 + 20y^2 = 200~\implies~\dfrac{x^2}{20} + \dfrac{y^2}{10}=1$

Therefore $a^2 = 20~\implies~a=\sqrt{20}=2\sqrt{5}$

The complete axis has the length $2a = 2\cdot 2\sqrt{5} = 4 \cdot \sqrt{5}$

3. Originally Posted by fabxx
heyy, (Barron SAT Math II Page 19)
what is the length of the major axis of the ellipse whose equation is 10x^2+20y^2=200?

a) 3.16
b)4.47
c) 6.32
d) 8.94
e) 14.14

The correct answer is D). The answer book said to divide both sides of the equation by 200 to write the equation in standard form. So then the length of the major axis is 2 $sqrt20$. Can please explain why? What does it mean by 'major' axis ?

The standard for of the equation of an ellipse is:

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

where the long axis is oriented along the $x$-axis ond the short along the $y$-axis.

The major axis is the long axis, and the minor axis is the short axis. $a$ is half the length of the major axis and $b$ half the length of the minor axis.

CB

4. Thanks for the answers. But how do you know that a is only half the axis? Thanks in advance

5. Originally Posted by fabxx
Thanks for the answers. But how do you know that a is only half the axis? Thanks in advance
Because when $y=0$:

$x^2=a^2$

so:

$x=\pm a$

are the two points where the ellipse cuts the $x$ axis, the distance between them $2a$, is the length of the major axis.

CB