# FINDING THE EQUATION OF A CIRCLE IN GENERAL FORM

## About "Finding the Equation of a Circle in General Form"

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

x2 + y2 = r2

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

(x - h)2 + (y - k) =  r2

x2 + y2 + 2gx + 2fy + c = 0

## Finding the Equation of a Circle in General Form - Practice questions

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) =  r2 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)2 + (y - k)2  =  r2(x - 0)2 + (y - 5)2  =  52x2 + y2 - 10y + 25 - 25  = 0x2 + y2 - 10y  = 0 Center (0, -5) and r = 5(x - h)2 + (y - k)2  =  r2(x - 0)2 + (y + 5)2  =  52x2 + y2 + 10y + 25 - 25  = 0x2 + y2 + 10y  = 0

Hence the required equations are x2 + y2 - 10y  =  0 and x2 + y2 + 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) =  r---(1)

(h, k)  ==>  (2, -1)

(x, y)  ==>  (3, 6)

(3 - 2)2 + (6 + 1) =  r2

12 + 7 =  r2

r=  50

(x - 2)2 + (y + 1) =  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) =  r2

(-4 + r)2 + (-2 + r) =  r2

16 + r2 - 8r + 4 - 4r + r2 - r2  =  0

20 + r2 - 12r  =  0

r2 - 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) =  102

x2 + 20x + 100 + y2 + 20y + 100 - 100  =   0

x2 + 20x + y2 + 20y + 100  =  0

x2 + y2 + 20x + 20y + 100  =  0

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

(x + 2)2 + (y + 2) =  22

x2 + 4x + 4 + y2 + 4y + 4 - 4  =   0

x2 + 4x + y2 + 4y + 4  =  0

x2 + y2 + 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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