Circle (Math)?

Let 0(a, 0) the coordinates of the center of the circle.

OA² = OB²

(a + 1)² + (0 - 1)² = (a - 1)² + (0 - 3)²

a² + 2a + 1 + 1 = a² - 2a + 1 + 9

4a = 8

a = 2

Radius of the circle, r

= √[(2 + 1)² + (0 - 1)²]

= √10

Equation of the circle:

(x - 2)² + y² = 10

x² + y² - 4x - 6 = 0

A(-1, 1), B(1, 3)

This would be a good problem for practice with general form. It is a useful thing to learn, but it might be less easy to follow. Notice that nowhere do I derive the coordinates of any point, the slope of any line, or the radius of any circle.

This is the circle having diameter AB:

(x + 1)(x - 1) + (y - 1)(y - 3) = 0

x² + y² - 4y + 2 = 0

This is line AB:

(1 - 3)(x - 1) - (-1 - 1)(y - 3) = 0

-2x + 2y - 4 = 0

x - y + 2 = 0

Any circle through A and B may be expressed in this form for some real k:

x² + y² - 4y + 2 + k(x - y + 2) = 0

x² + y² + kx + (-k - 4)y + 2k + 2 = 0

The center is on the x-axis only if the coefficient of y is zero.

-k - 4 = 0

k = -4

x² + y² + (-4)x + [-(-4) - 4]y + 2(-4) + 2 = 0

x² + y² - 4x - 6 = 0

The typical equation of a circle is: (x - xo)² + (y - yo)² = R² → where:

xo: abscissa of center → (xM) in your case

yo: ordinate of center → 0 in your case (because the center is on the x-axis

R: radius of circle

The equation of your circle is: (x - xo)² + y² = R²

The circle passes through point A (- 1 ; 1), so these coordinates must verify the equation of the circle

(x - xo)² + y² = R² → when: x = - 1, then: y = 1

(- 1 - xo)² + (1)² = R² ← memorize this result

The circle passes through point B (1 ; 3), so these coordinates must verify the equation of the circle

(x - xo)² + y² = R² → when: x = 1, then: y = 3

(1 - xo)² + (3)² = R² → recall them emorized result: (- 1 - xo)² + (1)² = R²

(1 - xo)² + (3)² = (- 1 - xo)² + (1)²

1 - 2xo + xo² + 9 = 1 + 2xo + xo² + 1

- 2xo + 9 = 1 + 2xo

- 4xo = - 8

xo = 2

Recall them emorized result: (- 1 - xo)² + (1)² = R²

R² = (- 1 - xo)² + (1)² → we've just seen that: xo = 2

R² = (- 1 - 2)² + (1)²

R² = 10

The equation of your circle is: (x - 2)² + y² = 10

Center is at: (x, 0)

distance from center to A = distance from center to B [radius]

(x + 1)² + 1 = (x - 1)² + 9

solve...easy...do something !!

x = 2

center is at: (2, 0)

now you can calculate the radius [√10]

(x - 2)² + y² = 10

Points A(-1,1) and B(1,3) are on a circle with equation of the form

r^2 = (x – a)^2 + y^2, and equating both forms for r^2

(-1 – a)^2 + 1 = (1 – a)^2 + 9

a^2 + 2a + 2 = a^2 - 2a + 10, so a = 2 and r^2 = 10

(x – 2)^2 + y^2 = 10