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What is the equation of the circle of radius 10 and center four, negative seven.
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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.
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And the radius is equal to 𝑟.
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In this case, we have a radius of 10 and a center of four, negative seven.
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Substituting these values into the equation gives us 𝑥 minus four all squared plus 𝑦 plus seven all squared equals 10 squared.
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The 𝑦 plus seven comes from the fact that 𝑦 minus negative seven is the same as 𝑦 plus seven.
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Squaring 𝑥 minus four is the same as multiplying 𝑥 minus four by 𝑥 minus four.
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We can expand these brackets, or parentheses, using the FOIL method.
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Multiplying the first terms, 𝑥 multiplied by 𝑥 gives us 𝑥 squared.
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Multiplying the outside terms gives us negative four 𝑥.
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Multiplying the inside terms also gives us negative four 𝑥.
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And finally, multiplying the last terms, negative four multiplied by negative four gives us positive 16.
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This means that 𝑥 minus four squared is the same as 𝑥 squared minus eight 𝑥 plus 16.
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We can expand the second bracket, 𝑦 plus seven all squared, in the same way.
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𝑦 plus seven all squared is equal to 𝑦 squared plus 14𝑦 plus 49.
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On the right-hand side, 10 squared is equal to 100.
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Grouping positive 16 and positive 49 gives us 𝑥 squared plus 𝑦 squared minus eight 𝑥 plus 14𝑦 plus 65 is equal to 100.
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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.
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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.