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.
|
|
|
