Question

1.Show that the points (5,5),(6,4),(-2,4) and (7,1) are concyclic .Also find the equation ,centre and radius of the circle on which they lie.​

Answer 1

Answer:

• See below

Step-by-step explanation:

This is how I solved it.

Plotted the given points. See the attached.

Identified two pairs to connect, the two chords.

• (5. 5) with (7, 1) and (6,4) with (-2, 4).

Determined the equation of both lines:

• y -5 = [( 1 - 5)/(7-5)](x - 5) ⇒  
• y - 5 = -2(x - 5)
• y = -2x + 15

and

• y = 4 (horizontal line)

Identified perpendicular bisectors of those two chords, they both pass through the center of circle.

Midpoints of the chords:

• [(5 + 7)/2, (5 + 1)/2] = (6, 3)

and

• (6 - 2)/ 2 = 2

The equations:

• y - 3 = 1/2(x - 6) ⇒ y = 1/2x
• x = 2 (perpendicular to y = 4 and passing through x = 2)

Solving the system found the intersecting of those, the center.

Found the center:

• (2, 1)

Found the distance from the center to one of the points (7, 1):

• 7 - 2 = 5, this is the radius

The equation of circle:

• (x - 2)² + (y - 1)² = 25

Verified all the other points are on same circle, confirmed.