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

Question Video: Finding the Equation of a Circle given Its Radius and Center Mathematics • Second Year of Secondary School

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Write in the form 𝑎𝑥² + 𝑏𝑦² + 𝑐𝑥 + 𝑑𝑦 + 𝑒 = 0, the equation of the circle of radius 10 and center (4, −7).

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Video Transcript

Write in the form 𝑎𝑥 squared plus 𝑏𝑦 squared plus 𝑐𝑥 plus 𝑑𝑦 plus 𝑒 equals zero the equation of the circle of radius 10 and center four, negative seven.

We begin by recalling that the general equation of any circle is given by 𝑥 minus 𝑝 all squared plus 𝑦 minus 𝑞 all squared is equal to 𝑟 squared, where the circle has center with coordinates 𝑝, 𝑞 and radius 𝑟. In this question, we are told that the radius is equal to 10 and the center of the circle is at the point four, negative seven. This means that 𝑝 is equal to four, 𝑞 is equal to negative seven, and 𝑟 is equal to 10. Substituting these values into the general equation, we have 𝑥 minus four all squared plus 𝑦 minus negative seven all squared is equal to 10 squared, which can be rewritten as 𝑥 minus four all squared plus 𝑦 plus seven all squared is equal to 10 squared.

We are asked to write the equation in the form 𝑎𝑥 squared plus 𝑏𝑦 squared plus 𝑐𝑥 plus 𝑑𝑦 plus 𝑒 is equal to zero. In order to do this, we will look at how we can rewrite 𝑥 minus four all squared and 𝑦 plus seven all squared. Using the FOIL method to distribute our parentheses, 𝑥 minus four all squared is equal to 𝑥 squared minus four 𝑥 minus four 𝑥 plus 16. In the same way, we see that 𝑦 plus seven all squared is equal to 𝑦 squared plus seven 𝑦 plus seven 𝑦 plus 49. By collecting like terms, the left-hand side of our equation of the circle is 𝑥 squared minus eight 𝑥 plus 16 plus 𝑦 squared plus 14𝑦 plus 49. This is equal to 10 squared, or 100.

We can then begin writing the terms in the order required. We have 𝑥 squared, 𝑦 squared, negative eight 𝑥, and 14𝑦. Subtracting 100 from both sides of our equation, on the left-hand side, we have 16 plus 49 minus 100. This is equal to negative 35. And our equation becomes 𝑥 squared plus 𝑦 squared minus eight 𝑥 plus 14𝑦 minus 35 is equal to zero. This is the equation of the circle of radius 10 and center four, negative seven in the required form.

The values of 𝑎, 𝑏, 𝑐, 𝑑, and 𝑒, respectively, are one, one, negative eight, 14, and negative 35. And the equation of the circle is 𝑥 squared plus 𝑦 squared minus eight 𝑥 plus 14𝑦 minus 35 equals zero.

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