Andrew M. answered • 10/02/17

Mathematics - Algebra a Specialty / F.I.T. Grad - B.S. w/Honors

Circle centered at (h,k) with radius r is (x-h)

^{2}+(y-k)^{2}= r^{2}a circle of a radius 3 centered at the origin:

x

^{2}+ y^{2}= 9 {equation 1}y = 4-2x {equation 2}

We have a circle and a line. We need to find the points of

intersection and find the distance between those two points.

Replace y with 4-2x in the 1st equation and solve for x.

x

^{2}+ (4-2x)^{2}= 9x

^{2}+ 16 - 16x + 4x^{2}- 9 = 05x

^{2}- 16x + 7 = 0Using quadratic equation with a=5, b=-16, c=7

x = [16 ±√((-16)

^{2}-4(5)(7))]/2(5)x = (16 ±√116)/10

x = (16 ± 10.77033)/10

x = 26.77033/10 = 2.677033

or

x = 5.22967/10 = 0.522967

We have the 2 values of x where the line intersects the circle.

Plug those into one of the original equations to find the associated

y values. Let's use y = 4-2x

For x = 2.677033

y = 4 - 2(2.677033) = -1.354066

**Intersection point (2.677033, -1.354066)**

for x = 0.522967

y = 4 - 2(0.522967) = 2.954066

**Intersection point (0.522967, 2.954066)**

Using the distance formula

d from (x

_{1}, y_{1}) to (x_{2}, y_{2}) = √[(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}]determine the distance between the two intersection points.

This will be the length of the line segment that lies within

the circle.

d = √[(2.677033-0.522967)

^{2}+(-1.354066-2.954066)^{2}]= √(4.64000033 + 18.56000133)

= √23.2000166

= 4.816638

You can round to as many decimal places as needed

d ≅ 4.82