Question

How many points of intersection are there for x-y=2 and x^2+y^2=25 ?

Answer 1

A circle is the set of all points that are the same distance, r, from a fixed point.

General Formula:

x ^ 2 + y ^ 2 = r ^ 2

where r is the radius

In this case :

x ^ 2 + y ^ 2 = 25 = 5 ^ 2

The circle has a center of (0,0) and a radius of 5


x - y = 2

x - 2 = y


y = x - 2


The straight line.



Points of intersection:

x ^ 2 + y ^ 2 = 25

Substitute y = x - 2

x ^ 2 + ( x - 2 ) ^ 2 = 25

x ^ 2 + ( x ^ 2 - 2 * x * 2 + 2 ^ 2 ) = 25

x ^ 2 + ( x ^ 2 - 4 * x + 4 ) = 25

x ^ 2 + x ^ 2 - 4 * x + 4 = 25

2 x ^ 2 - 4 x + 4 - 25 = 0

2 x ^ 2 - 4 x - 21 = 0


If you don't know how to solve this equation in google type:

quadratic equation online

When you see list of results click on:

Free Online Quadratic Equation Solver:Solve by Quadratic Formula

When page be open in rectangle type:

2 x ^ 2 - 4 x - 21 = 0

and click option: solve it

You will see solution step-by step


So solutions are :


x = 1 + sqrt ( 23 / 2 ) = 4.39116


y = x - 2 = 1 + sqrt ( 23 / 2 ) - 2 =

sqrt ( 23 / 2 ) - 1 = 2.39116


and


x = 1 - sqrt ( 23 / 2 ) = - 2.39116


y = x - 2 = 1 - sqrt ( 23 / 2 ) - 2 =

- 1 - sqrt ( 23 / 2 ) - 1 = - 4.39116



If you want to see graph in google type:

functions graphs online

When you see list of results click on:

rechneronline.de/function-graphs/

When page be open in blue recatangle type:

(25-x^2)^0.5

In gray recatangle type:

-(25-x^2)^0.5

In white recatangle type:

x-2

And click option :

Draw


You will see 2 points of intersection.