math question please show work,?

Question

givin the circle x^2+ y^2 = 20, write the equation of any line such that ..
a)the line and circle form a system with two differnt solutions
b) the line and the circle form a system with no solutions
hint: x^2+y^2 is a circle w/ a center @ (0,0) abd radius = sqaure root of 20

Answer 1

Using the equation for a circle and the equation for a line as a system of equations, yields one solution for each point where the line intersects the circle.

If you are asked for a system with two solutions, you have to find a line that passes through the circle, crossing the circle in two places. If you are asked for a system with no solutions, you have to find a line that never touches the circle. (If you were aksed to find a system with one solution, you'd have to find a line tangent to the circle -- that would be the hardest one to answer.)

(a) You are given that the circle is centered on the origin. Any line which passes through the origin will intersect the circle in two different places, so the first part is easy.

For example, the line x=y two solutions:

x^2+ y^2 = 20
x^2 + x^2 = 20 (substitute for x=y)
2x^2 = 20
x^2 = 10
x = +/- sqrt(10)

Since y=x, the two points
(sqrt(10), sqrt(10)) and
(-sqrt(10), -sqrt(10)) are solutions.

So does the line x=2:

x^2 + y^2 = 20
2^2 + y^2 = 20
y^2 = 16
y = +/- 4

The solutions for that system are (2,4) and (2,-4).

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(b) is a little harder. A circle with radius sqrt(20) (which is between 4 and 5) centered on the origin will not have any x-values or y-values greater than 5 (or less than -5).

The line x=5, for example, does not intersect the circle at all, and therefore has no solutions when used in combination with the circle's equation as a system of equations:

x^2 + y^2 = 20
5^2 + y^2 = 20 (substitute for x=5)
y^2 = -5

That equation (y^2 = -5) has no solutions in the real numbers.

Answer 2

thats a good question! thanks for the 2 points!!!!!!!!!!1