Question

Topic : Circles
If (2,0) , (0,1) , (4,5) & (0,c) are concyclic then find c.
if you answer i will make your answer as brainalist answer...​

Answer 1

Answer:

→Let the equation of the required circle,

x²+y²+2gx+2fy+c=0→1

→Since equation 1 is passing through the point (2,0)

2²+0²+4g+0f+c=0

4g+c=-4→2

→Since equation 1 is passing through the point (0,1)

0²+1²+0g+2f+c=0

2f+c=-1→3

→Since equation 1 is passing through the point (4,5)

4²+5²+8g+10f+c=0

8g+10f+c=-41→4

From the equations,2 and 3

4g+c=-4

2f+c=-1

on solving, 4g-2f=-3→5

From the equations,3 and 4

2g+c=-1

8g+10f+c=-41

on solving,-g-f=5→6

Now, solve equation 5 and 6

g f 1

-2 3 4 -2

-1 -5 -1 -1

g/10+3=f/-3+20=1/-6

g=-13/6, f=-17/6

Substitute the values of g,f in equation 2

4g+c=-4

4(-13/6)+c=-4

3c-26=-12

3c=14

c=14/3

Therefore, Equation of the circle is,

x²+y²+2x(-13/6)+2y(-17/6)+14/3=0

3x²+3y²-13x-17y+14=0

Given four points are concyclic

(0,C) lies on the circle.

3×0²+3(c²)-13×0-17c+14=0

3c²-17c+14=0

3c²-3c-14c+14=0

3c(c-1)-14(c-1)=0

c=1, c=14/3

Hope it helps you frnd.......