Work each of the following. Find the equation of a circle with...

Question

Work each of the following. Find the equation of a circle with center at (-4, 3), passing through the point (5, 8).Write it in center-radius form.

Accepted Answer

Hey, everyone here we are told that we have a certain circle with the center at negative two and nine. And it passes through the 90.7 and 13. We are asked to write the equation that best describes the circle in center radius form. So here we have four answer choice options. A B C and D each with a slightly different equation that represents our circle. So we want to begin this problem by first identifying our given information. Our first piece of information is the center of the circle. Well, we need to recall that the center of the circle is the point H K. So the point H K is the point negative two and nine. And next, we are told that the circle passes through the 90.7 and 13. So just a regular X and Y value is the 130. and 13. So we are also asked to write the equation in center radius form. So we need to recall the formula for this circle and center radius form which is the quantity of X minus H squared plus the quantity of Y minus K squared is equal to R squared. So now we need to find our radius R so that we can use this formula to write our equation of the circle. So we can find the radius R by plugging in all of our given values into this formula here. So we have the quantity of X which is seven minus H which is negative two. And this quantity is squared, then we have plus the quantity of Y which is 13 minus K which is nine. And this quantity is also squared and this expression is equal to R squared according to our formula. So now we just need to start simplifying. So first, we have seven minus negative two, which is the same as seven plus two. So we have the quantity of seven plus two square plus the quantity of 13 minus nine, which is four and this is squared and this expression is still equal to R squared. Now continuing to simplify seven plus two gives us nine. So we have nine squared plus four squared is equal to R squared. Now squaring both of our terms, we have 81 plus 16 is equal to R squared. Adding the terms together, we have 97 is equal to R squared. Now, we need to get rid of the squared value for our, so we need to take the square root of both sides and doing this, we see that our value for our is the square root of 97. So now that we have found the radius of our circle. We can go ahead and use our formula from earlier to write the equation of our circle. We just need to exclude the point given for the X and Y values. So doing this, we have the quantity of X minus H which is negative two. So we have minus negative two. And this quantity is squared plus the quantity of Y -K which is nine. And this quantity is also squared and this is equal to R squared. So this is equal to the quantity of the square root of 97 squared. So now we just need to simplify our equation here. So X minus negative two is the same as X plus two. So we have the quantity of X plus two squared plus the quantity of Y minus nine squared is equal to. And now the square root of 97 quantity squared is just 97. So now we have simplified our equation and again, we have the quantity of X plus two squared plus the quantity of Y minus nine squared is equal to 97. So with this equation, we are left with answer choice. A thanks for watching. I hope you found this video helpful and I'll see you again next time.

Work each of the following. Find the equation of a circle with center at (-4, 3), passing through the point (5, 8).Write it in center-radius form.

Key Concepts

Equation of a Circle in Center-Radius Form

Distance Formula

Substitution into the Circle Equation

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