Real Solutions (x,y)

How  many real solutions (x,y) are there that satisfy the two equations x^2+y^2=30 and 4y^2-x^2=100?
Source: NCTM Mathematics Teacher, September 2006

Solution
image
The graph of x^2+y^2=30 is a circle centered at the origin and radius = \sqrt{30}. The graph of 4y^2-x^2=100 is a hyperbola with vertices at (0,5) and (0,-5). The two graphs intersect at four points. There are four real solutions (x,y) that satisfy the two equations.

Answer: 4

Alternate solution
Add the two equations
x^2+y^2=30
4y^2-x^2=100
————————
5y^2=130
y^2=26
y=\pm\sqrt{26}
Substitute the value of y^2 into the first equation
x^2+26=30
x^2=4
x=\pm 2

Advertisements

About mvtrinh

Retired high school math teacher.
This entry was posted in Problem solving and tagged , , , , , , , . Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google+ photo

You are commenting using your Google+ account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

w

Connecting to %s