SOLUTION: The endpoint of the radius of a circle with centre C (4,1) is D (1,6). Determine the coordinates of the endpoint E where DE is the diameter of the circle.

Algebra ->  Coordinate-system -> SOLUTION: The endpoint of the radius of a circle with centre C (4,1) is D (1,6). Determine the coordinates of the endpoint E where DE is the diameter of the circle.       Log On


   

Question 1175478: The endpoint of the radius of a circle with centre C (4,1) is D (1,6).
Determine the coordinates of the endpoint E where DE is the diameter of the circle.


Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
given:
centre C (4,1) -> h=4, k=1
distance between C and D is equal to the radius
r%5E2=%281-4%29%5E2%2B%286-1%29%5E2..eq.1
r%5E2=9%2B25
r%5E2=34
formula of the circle is:
%28x-h%29%5E2%2B%28y-k%29%5E2=r%5E2
%28x-4%29%5E2%2B%28y-1%29%5E2=34..........eq.1
the endpoint of the radius of a circle is D (1,6)
DE is the diameter of the circle, so find equation of the line passing through the endpoint of the radius D (1,6) and centre C (4,1)
y=mx%2Bb
use points to find a slope:
m=%281-6%29%2F%284-1%29=-5%2F3
so far equation is
y=-%285%2F3%29x%2Bb+.....use one point to calculate b
1=-%285%2F3%294%2Bb
1=-20%2F3%2Bb
b=1%2B20%2F3
b=23%2F3
and equation of the line is
y=-%285%2F3%29x%2B23%2F3..........eq.2
the endpoint E (x,y) will be intersection point of the circle and line
so, to find it solve this system
%28x-4%29%5E2%2B%28y-1%29%5E2=34..........eq.1
y=-%285%2F3%29x%2B23%2F3..........eq.2
-------------------------------------
substitute y from eq.2 in eq.1 and solve for x
%28x-4%29%5E2%2B%28-%285%2F3%29x%2B23%2F3-1%29%5E2=34
%28x-4%29%5E2%2B%28-%285%2F3%29x%2B23%2F3-1%29%5E2=34
34%2F9+%28x+-+4%29%5E2+=+34
%28x+-+4%29%5E2+=+34%2A9%2F34
%28x+-+4%29%5E2+=+9
%28x+-+4%29=3 or %28x+-+4%29=-3
solutions:
x+=7+or x+=1
go to
y=-%285%2F3%29x%2B23%2F3..........eq.2, substitute x
if x=7
y=-%285%2F3%297%2B23%2F3
y=-4

if x=1
y=-%285%2F3%291%2B23%2F3
y=6
intersection points are: (7,-4) and (1,6)
since given that D (1,6), then E (7,-4)




Answer by ikleyn(52790) About Me  (Show Source):
You can put this solution on YOUR website!
.

            Hello, the solution is  MUCH  SIMPLER  (!). 


From the point D(1,6) to the center C(4,1),  you move 3 units horizontally in positive direction of x-axis;


    hence, to get the other diameter's endpoint, you CONTINUE to move another 3 units horizontally along x-axis

    from coordinate x= 4 (center) to x= 4+3 = 7 (endpoint).



Similarly,

from the point D(1,6) to the center C(4,1),  you move 5 units vertically in negative direction of y-axis;


    hence, to get the other diameter's endpoint, you CONTINUE to move another 5 units vertically along y-axis

    from coordinate y= 1 (center) to y= 1-5 = -4 (endpoint).



Thus, the other midpoint coordinates are  x= 7,  y =-4,  and the endpoint itself is (7,-4).     ANSWER


That is all.


YOU  DO  NOT  NEED  solve any complicated equations to solve this  SUPER-simple problem  (!)


You do not need write this  Master's thesis that another tutor impose you  (!)


May the  LORD  saves you of doing this  UNNECESSARY  work  (!)


M E M O R I Z E   this algorithm as a mantra and use it  EVERY  TIME  when you solve similar problems  (!)


On a test,  normal time to complete such assignment is  10 - 15  seconds.