Video: Finding the Equation of a Circle given Its Radius and Center

What is the equation of the circle of radius 10 and center (4, −7).

02:45

Video Transcript

What is the equation of the circle of radius 10 and center four, negative seven.

The general equation of a circle is given by 𝑥 minus 𝑎 all squared plus 𝑦 minus 𝑏 all squared is equal to 𝑟 squared, where the center of the circle has coordinates 𝑎, 𝑏. And the radius is equal to 𝑟. In this case, we have a radius of 10 and a center of four, negative seven. Substituting these values into the equation gives us 𝑥 minus four all squared plus 𝑦 plus seven all squared equals 10 squared. The 𝑦 plus seven comes from the fact that 𝑦 minus negative seven is the same as 𝑦 plus seven. Squaring 𝑥 minus four is the same as multiplying 𝑥 minus four by 𝑥 minus four.

We can expand these brackets, or parentheses, using the FOIL method. Multiplying the first terms, 𝑥 multiplied by 𝑥 gives us 𝑥 squared. Multiplying the outside terms gives us negative four 𝑥. Multiplying the inside terms also gives us negative four 𝑥. And finally, multiplying the last terms, negative four multiplied by negative four gives us positive 16. This means that 𝑥 minus four squared is the same as 𝑥 squared minus eight 𝑥 plus 16.

We can expand the second bracket, 𝑦 plus seven all squared, in the same way. 𝑦 plus seven all squared is equal to 𝑦 squared plus 14𝑦 plus 49. On the right-hand side, 10 squared is equal to 100. Grouping positive 16 and positive 49 gives us 𝑥 squared plus 𝑦 squared minus eight 𝑥 plus 14𝑦 plus 65 is equal to 100. And subtracting 100 from both sides give us a final equation of 𝑥 squared plus 𝑦 squared minus eight 𝑥 plus 14𝑦 minus 35 equals zero.

Therefore, the equation of the circle with radius 10 and center four, negative seven is 𝑥 squared plus 𝑦 squared minus eight 𝑥 plus 14𝑦 minus 35 equals zero.

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