Question

Show that the points (5, 5), (6, 4) (-2, 4) and (7, 1) are concyclic. Find the equation, centre and radius of this circle.

Answer 1

Answer:

Equation of Circle is $(x-2)^2+(y-1)^2 = 25$

Centre of circle is ( 2, 1 )

Radius of circle is 5 unit

Step-by-step explanation:

Given four points are ( 5, 5 ), ( 6, 4 ), ( -2, 4 ) & ( 7, 1 )

To check these points are concyclic we find equation of circle using three point form and check if 4th point satisfies it.

General Form of circle given by $x^2+y^2+Dx+Ey+F=0$

we find value of variable D, E, F by putting value of points  ( 5, 5 ), ( 6, 4 ), ( -2, 4 )

from ( 5, 5 ) we get $5^2+5^2+5D+5E+F=0$

⇒ 5D + 5E + F + 50 = 0......................................(eqn 1)

from ( , 5 ) we get $6^2+4^2+6D+4E+F=0$

⇒ 6D + 4E + F + 52 = 0......................................(eqn 2)

from ( 5, 5 ) we get $(-2)^2+4^2+(-2)D+4E+F=0$

⇒ -2D + 4E + F + 20 = 0......................................(eqn 3)

Now By simplifying these 3 equation we get values of D, E, F

subtracting eqn 1 from 2 we get, D - E + 2 = 0................(eqn 4)

subtracting eqn 3 from 2 we get, 8D + 32 = 0 ⇒ D = $\frac{-32}{8}$ ⇒ D = -4

putting this value in eqn 4 we get, -4 - E + 2 = 0 ⇒ E = -4 + 2 ⇒ E = -2

by substitue value of D and E in eqn 1 we get, 5(-4) + 5(-2) + F + 50 = 0

⇒ -20 -10 + F + 50 = 0

⇒ F = -50 + 30 ⇒ F = -20

∴ The general form of circle is $x^2+y^2-4x-2y-20=0$

Now substituing coordinates of 4th point, we get

$LHS = x^2+y^2-4x-2y-20\\= 7^2+1^2-4\times7-2\times1-20\\=49+1-28-2-20\\=0=RHS$

Therefore, Given Points are concyclic.

Standard equation of circle is $(x-h)^2+(y-k)^2=r^2$

where ( h, k ) is coordinate of centre circle and r is radius of circle.

Now we convert General form in Standard Equation of circle using Completing the square method.

$x^2+y^2-4x-2y-20=0\\ (2^2\:and\: 1^2\: is\: added \:and\: subtracted\: to\: complete\: the\: square)\\\implies x^2+y^2-4x-2y+2^2-2^2+1^2-1^2-20=0\\\implies (x^2-4x+2^2)+(y^2-2y+1^2)-2^2-1^2-20=0\\\implies (x-2)^2+(y-1)^2-4-1-20 = 0\\\implies (x-2)^2+(y-1)^2 = 25\\\implies (x-2)^2+(y-1)^2 = 5^2$

By comparing with standard equation,

Coordinate of centre of circle is ( 2, 1 ) and radius of circle is 5 unit.