How to find the center and radius of a circle given an equation
The general equation of a circle is thus in the form x² + y² + 2ex + 2fy + c = 0;
I learned it during my first semester and year in my undergraduate degree.
Let's say for example you are asked to find the centres and radii or radius of a given circle below
x² + y² - 6x - 8y + 5 = 0
Rule number 1: The coefficient of x² and y² must be 1 when determining and solving the center and radius of a circle in an equation and the 1 can be visible or invisible.
And in any given example, you must apply the general formula to the equation given to you
x² + y² - 6x - 8y + 5 = 0
x² + y² + 2ex + 2fy + 5 = 0 In this case, we have to look for the value of e and f and to do so
You have solve the equation 2ex = -6x and 2fy = -8y and to do so, we have to apply the law of transposition and division which is to divide both side of the equation by 2x and 2y to the applicable equation and if we do that. Our final value will be
e = -3 because -6 divided by 2 is -3
f = -4 because -8 divided by 2 is -4
Rule Number 2: whatever you divide by two in the equation, the final value must be positive or negative. That is to say our centre a and b are described below
a = -e In this case our e is -3 and going by the law of BODMAS in which numbers in brackets have to be evaluated first and also
negative * negative = positive
Positive * negative = negative and so our calculation is -(-3) = 3
b = -f is also -(-4) equal to - * -4 = 4
And so our centre a, b is 3 and 4 respectively.
To find the radius of a circle given an equation. The formula is
Note: c represent a constant
r = square root of √a² + b² - c
To apply this to our example. We will have 3² + 4² - 5 which is equal or equivalent to
9 + 16 - 5 and the evaluation starts from left to right as both addition and subtraction have a precedence of 1. Our final answer will be 20 and the square root of 20 is 2√5
20 is not a perfect square, so we have to find two numbers which multiply together to give the product of 20 and out of the two numbers we look for the square of one of the number and for example 4 * 5. The square root of 4 is 2 and so the answer is 2√5.
I learned it during my first semester and year in my undergraduate degree.
Let's say for example you are asked to find the centres and radii or radius of a given circle below
x² + y² - 6x - 8y + 5 = 0
Rule number 1: The coefficient of x² and y² must be 1 when determining and solving the center and radius of a circle in an equation and the 1 can be visible or invisible.
And in any given example, you must apply the general formula to the equation given to you
x² + y² - 6x - 8y + 5 = 0
x² + y² + 2ex + 2fy + 5 = 0 In this case, we have to look for the value of e and f and to do so
You have solve the equation 2ex = -6x and 2fy = -8y and to do so, we have to apply the law of transposition and division which is to divide both side of the equation by 2x and 2y to the applicable equation and if we do that. Our final value will be
e = -3 because -6 divided by 2 is -3
f = -4 because -8 divided by 2 is -4
Rule Number 2: whatever you divide by two in the equation, the final value must be positive or negative. That is to say our centre a and b are described below
a = -e In this case our e is -3 and going by the law of BODMAS in which numbers in brackets have to be evaluated first and also
negative * negative = positive
Positive * negative = negative and so our calculation is -(-3) = 3
b = -f is also -(-4) equal to - * -4 = 4
And so our centre a, b is 3 and 4 respectively.
To find the radius of a circle given an equation. The formula is
Note: c represent a constant
r = square root of √a² + b² - c
To apply this to our example. We will have 3² + 4² - 5 which is equal or equivalent to
9 + 16 - 5 and the evaluation starts from left to right as both addition and subtraction have a precedence of 1. Our final answer will be 20 and the square root of 20 is 2√5
20 is not a perfect square, so we have to find two numbers which multiply together to give the product of 20 and out of the two numbers we look for the square of one of the number and for example 4 * 5. The square root of 4 is 2 and so the answer is 2√5.