**Finding the Equation of a Circle in General Form :**

Here we are going to see some example problems to know how to find equation of a circle in general form.

Equation of the circle with centre (0,0) and radius r

x^{2} + y^{2} = r^{2}

Equation of circle with centre (h, k) and radius r

(x - h)^{2} + (y - k)^{2 } = r^{2}

x^{2} + y^{2} + 2gx + 2fy + c = 0

**Question 1 :**

Obtain the equation of the circles with radius 5 cm and touching x-axis at the origin in general form.

**Solution :**

General form of a circle :

(x - h)^{2} + (y - k)^{2 } = r^{2}

The center lies on y axis. The center point will be at (0, 5) and (0, -5).

Center (0, 5) and r = 5 (x - h) (x - 0) x x |
Center (0, -5) and r = 5 (x - h) (x - 0) x x |

Hence the required equations are x^{2} + y^{2} - 10y = 0 and x^{2} + y^{2} + 10y = 0.

**Question 2 :**

Find the equation of the circle with centre (2,-1) and passing through the point (3,6) in standard form.

**Solution :**

(x - h)^{2} + (y - k)^{2 } = r^{2 }---(1)

(h, k) ==> (2, -1)

(x, y) ==> (3, 6)

(3 - 2)^{2} + (6 + 1)^{2 } = r^{2}

1^{2} + 7^{2 } = r^{2}

r^{2 }= 50

(x - 2)^{2} + (y + 1)^{2 } = 50

**Question 3 :**

Find the equation of circles that touch both the axes and pass through (-4,-2) in general form

**Solution :**

The center point will be at (-r, -r)

By applying the point passes through the circle and center, we get

(x - h)^{2} + (y - k)^{2 } = r^{2 }

(-4 + r)^{2} + (-2 + r)^{2 } = r^{2 }

16 + r^{2} - 8r + 4 - 4r + r^{2} - r^{2} = 0

20 + r^{2} - 12r = 0

r^{2} - 12r + 20 = 0

(r - 10) (r - 2) = 0

r = 10 and r = 2

Equation of a circle center is at (-10, -10) and radius is 10.

(x + 10)^{2} + (y + 10)^{2 } = 10^{2 }

x^{2} + 20x + 100 + y^{2} + 20y + 100 - 100 = 0

x^{2} + 20x + y^{2} + 20y + 100 = 0

x^{2} + y^{2} + 20x + 20y + 100 = 0

Equation of a circle center is at (-2, -2) and radius is 2.

(x + 2)^{2} + (y + 2)^{2 } = 2^{2 }

x^{2} + 4x + 4 + y^{2} + 4y + 4 - 4 = 0

x^{2} + 4x + y^{2} + 4y + 4 = 0

x^{2} + y^{2} + 4x + 4y + 4 = 0

After having gone through the stuff given above, we hope that the students would have understood, "Finding the Equation of a Circle in General Form".

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