|
Question 1158612: find the equation of the circle that passes through the 3 points (5, -7), (2, -4), and (5, -3).
Found 3 solutions by solver91311, MathLover1, greenestamps:
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
The key to this problem is the fact that the perpendicular bisector of any chord of a circle will pass through the center of the circle. Three non-colinear points, such as the three points you are given are the endpoints of three different chords of the desired circle. You only need to consider two of those chords.
Step 1: Choose any two of the three given points and use the midpoint formulas to find the coordinates of the chord defined by the two given endpoints. Hint: I would choose (5,-7) and (5,-3) as one of the pairs since these two points lie on a vertical line and this will greatly simplify your calculations going forward.
Step 2: Calculate the slope of the line that contains the two chosen points. Remember that a vertical line has an undefined slope and an equation of the form  some constant value.
Step 3: Calculate the negative reciprocal of the value you obtained in step 2. This is the slope of a perpendicular to the chord. A perpendicular to a vertical line is a horizontal line that has a slope of zero. I.e., an equation of the form  some constant value.
Step 4: Use the Point-Slope form of an equation of a straight line with the midpoint coordinates and the slope of the perpendicular found in steps 1 and 3 to write an equation of the perpendicular bisector of your first chord.
Step 5: Repeat Steps 1 through 4 using a different pair of the given points. You will then have equations of two perpendicular bisectors of two chords of the desired circle, which, as was noted earlier, intersect at the center of the circle.
Step 6: Use any convenient method to solve the 2X2 system of equations that resulted from steps 4 and 5. The unique solution will be the coordinates of the center of the desired circle.
Step 7: Use the distance formula to calculate the distance from the center to any one of the given points. This will be the radius of the desired circle. Hint: Don't actually take the square root as the final step of calculating the measure of the radius because your final result requires the radius squared.
Step 8: Write the equation of the circle:
^2\ +\ (y\ -\ k)^2\ =\ r^2)
Where  is the  -coordinate of the circle center,  is the  -coordinate of the circle center, and  is the radius of the circle. All three values being the results of your Step 6 and Step 7 calculations. Be careful with your signs -- if either coordinate is negative, then you retain that sign and end up with a plus (minus a minus) in the final expression.
Enjoy.
John
My calculator said it, I believe it, that settles it
Answer by MathLover1(20850)  (Show Source):
You can put this solution on YOUR website!
the equation of the circle:
if the circle passes through the 3 points ( ,  ), ( ,  ), and (,  ), we will use them to find  , , and 
.....plug in coordinates of the point (, )
 ......eq.1
.....plug in coordinates of the point (, )
 .......eq.2
.....plug in coordinates of the point (, )
 .....eq.3
subtract eq.2 from eq.1
 ......simplify
 .........simplify, both sides divide by 
 ..........solve for
 .......eq.1a
subtract eq.3 from eq.2
 ........simplify
 ......simplify
 ........solve for
 ......eq.2a
from eq.1a and eq.2a we have
 ......solve for
go to
......eq.2a, substitute
so, the center is at ( ,  )
plug it in
 ........now use the point (, ) to calculate
and, your equation is:
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
Two of the given points are (5,-7) and (5,-3). With the same x coordinate, the center of the circle (h,k) has to be on the line y=-5.
So we need to find h for which the distances from (h,-5) to (5,-3) and to (2,-4) are equal.
So the center of the circle is (h,k) = (4,-5).
The radius is the distance from (4,-5) to any one of the given points:
The equation is
|
|
|
| |
