SOLUTION: Find an equation for the line that goes through the two intersection points of the circle x2 +y2 =25andthecircle(x−8)2 +(y−4)2 =65.

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Question 1158991: Find an equation for the line that goes through the two intersection points of the circle
x2 +y2 =25andthecircle(x−8)2 +(y−4)2 =65.
Found 3 solutions by solver91311, ikleyn, MathTherapy:
Answer by solver91311(24713)   (Show Source): You can put this solution on YOUR website!


Conceptually, this is a straightforward problem. Step 1: Find the coordinates of the two points of intersection of the circles. Step 2: Use the Two-Point Form of an equation of a straight line to create the desired equation and put it into the desired form. However, step 1 requires some rather ugly algebra.

First we need the two circle equations expressed as functions of .





And





By setting the two right-hand sides equal to each other, we eliminate the variable .



Square both sides:



Simplify (I left out a lot of steps, but you can verify so that you can show your work if necessary)



Square both sides again:







Therefore the points of intersection are



Now all that you need to do is write a function describing the set of ordered pairs that comprise the line that contains these two points.

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I looked at my notes from this problem this morning, slapped my forehead, and got back into the system to edit my response. No need. Ikleyn and MathTherapy have it right.


John

My calculator said it, I believe it, that settles it
Answer by ikleyn(52803)   (Show Source): You can put this solution on YOUR website!
.

            Notice that the problem  DOES  NOT  ASK  YOU  to find the coordinates of the intersection points.

            It only ask you to find a linear equation for the line, which goes through these points.

            Therefore,  MUCH  SIMPLER  solution is possible. 


If the point (x,y) lies on both curves/(circles), given by their equations


    x^2 + y^2 = 25,            (1)

    (x-8)^2 + (y-4)^2 = 65,    (2)


then this point belongs to the intersection of these curves and satisfies to the equation 
which is the difference of equations (1) and (2).


The difference of these equations CANCEL the quadratic terms, leaving only linear terms

    x^2            + y^2           = 25     (1)

    x^2 - 16x + 64 + y^2 - 8y + 16 = 65     (2)

---------------------------------------------------  Take the difference eq(1) - eq(2)


          16x - 64       + 8y  - 16 = 25 - 65

          16x + 8y                  = 25 - 65 + 64 + 16

          16x + 8y                  = 40

           2x + y                   = 5.


ANSWER.  The equation, the problem asks for, is  2x + y = 5.

Solved.


Answer by MathTherapy(10553)   (Show Source): You can put this solution on YOUR website!
Find an equation for the line that goes through the two intersection points of the circle
x2 +y2 =25andthecircle(x−8)2 +(y−4)2 =65.
The points where the circles intersect each other are the solutions to the 2 equations. 

The 2 points of intersection are connected by a line, and that line is the equation that's being sought!

Therefore, we get: 

       Therefore, we get: 25 - 16x - 8y + 80 = 65 ------ Substituting 25 for x2 + y2 in eq (ii)

                             - 16x - 8y = - 40_____- 8(2x + y) = - 8(5) =====>  

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