SOLUTION: Part 1: Suppose you need to solve a system of equations in which one equation represents a circle and the other represents a line. How many solutions can your system have? Sele

Question 349538: Part 1:
Suppose you need to solve a system of equations in which one equation represents a circle and the other represents a line. How many solutions can your system have? Select ALL that are possible.
0,1,2,3,4, infinitely many
Part 2:
How many solutions does the following system of equations have?
x2+y2=64
x=1
0,1,2,3,4, infinitely many
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
one equation represents a circle and the other equation represents a line.

you can have 0, 1, or 2 solutions.

0 if the line and the circle never intersect
1 if the line is tangent to the circle.
2 if the line intersects the circle at any other angle.

the line can never intersect the circle at more than 2 points.

your equation is:

x^2 + y^2 = 64
x = 1

this is like solving equations simultaneously to get a common solution.

if x = 1, you can substitute in the first equation to get 1 + y^2 = 64 which give you y^2 = 63 which gives you y = +/- sqrt(63).

this implies 2 solutions to this equation.

The graph of the equation of the circle should confirm this.

To graph the equation of the circle, solve for y to get:

y = +/- sqrt(64-x^2)

that graph is shown below:

Draw a vertical line at x = 1 and you will see that the intersection of that vertical line and the circle is around y = +/- 7.937253933.

I drew 2 horizontal lines at y = +/- 7.937253933 so you can see that a little easier.

when x = 0, y = +/- sqrt(64-0) = +/- sqrt(64) = +/- 8.

That's the highest and lowest point of the circle.

when x = 1, y = +/- sqrt(64-1) = +/- sqrt(63) = +/- 7.937253933.

That's close to the highest point but hot exactly there (just a little below).

Solving the 2 equations simultaneously yielded the result.

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