SOLUTION: Consider the circle {{{x^2+y^2-6x-8y+9=0}}} and a secant line y=x. Determine the length of the corresponding secant segment.

Algebra ->  Points-lines-and-rays -> SOLUTION: Consider the circle {{{x^2+y^2-6x-8y+9=0}}} and a secant line y=x. Determine the length of the corresponding secant segment.      Log On


   

Question 1012679: Consider the circle x%5E2%2By%5E2-6x-8y%2B9=0 and a secant line y=x. Determine the length of the corresponding secant segment.
Found 2 solutions by Theo, macston:
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
you have two equations that need to be solved simultaneously.

the solution will be common to both equations.

the equations are:

x^2 + y^2 - 6x - 8y + 9 = 0

y = x

replace y with x in the first equataion to get x^2 + x^2 - 6x - 8x + 9 = 0

combine like terms to get 2x^2 - 14x + 9 = 0

solve this quadratic equation to get x = .71612 or x = 6.28388 rounded to 5 decimal places.

since y = x, the coordinate points are (.71612,.71612) or (6.28388,6.28388).

these points are the points of intersection between the circle and the secant line.

the formula for the length of that secant line is L = sqrt((x2-x1)^2 + (y2-y1)^2).

that becomes L = sqrt((6.28388-.71612)^2 + (6.28388-.71612)^2).

this results in L = 7.874 rounded to 3 decimal places.

you can also find the intersection points by graphing as shown below.

$$$




Answer by macston(5194) About Me  (Show Source):
You can put this solution on YOUR website!
STEP 1: Find intersection points of secant line and circle.
(solve simultaneous equations by substitution)
.
y=x
x%5E2%2By%5E2-6x-8y%2B9=0 . Substitute for y (y=x)
x%5E2%2By%5E2-6x-8x%2B9=0
2x%5E2-14x%2B9=0
.
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 2x%5E2%2B-14x%2B9+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-14%29%5E2-4%2A2%2A9=124.

Discriminant d=124 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--14%2B-sqrt%28+124+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-14%29%2Bsqrt%28+124+%29%29%2F2%5C2+=+6.28388218141501
x%5B2%5D+=+%28-%28-14%29-sqrt%28+124+%29%29%2F2%5C2+=+0.716117818584989

Quadratic expression 2x%5E2%2B-14x%2B9 can be factored:
2x%5E2%2B-14x%2B9+=+2%28x-6.28388218141501%29%2A%28x-0.716117818584989%29
Again, the answer is: 6.28388218141501, 0.716117818584989. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B-14%2Ax%2B9+%29

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x=6.28388218141501 --OR-- x=0.716117818584989
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Since y=x, the endpoints of the secant segment are:
(x%5B1%5D,y%5B1%5D)=(6.28388218141501, 6.28388218141501)
(x%5B2%5D,y%5B2%5D)=(0.716117818584989, 0.716117818584989)
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STEP 2: Find the distance between the points:
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distance=sqrt%28%28x%5B2%5D-x%5B1%5D%29%5E2%2B%28y%5B2%5D-y%5B1%5D%29%5E2%29
.
distance=sqrt%28%280.72-6.28%29%5E2%2B%280.72-6.28%29%5E2%29
.
distance=sqrt%282%28-5.56%29%5E2%29
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distance=sqrt%282%295.56=7.86( approx)
.

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I don't know why there is a gap in the circle. It shouldn't be there.