Explanation for Question 15 From the Math (No Calc) Section on the 2017 April Sat
Question 15 says the graph of the X squared minus four X plus Y 2 squared plus six Y minus 24 equals zero in the X, 3 Y plane is a circle. What is the radius of the circle? 4 So to figure out what the radius of a circle is from it's equation, 5 you first need to get it into the form of this equation. 6 I've written down here, X minus H squared, 7 plus Y minus K squared equals R square where R represents the radius. 8 So to do that, I'm just going to start by rewriting this equation, 9 but by grouping together like terms, so X squared minus four X plus 10 Y squared plus six Y, and then I'm going to add 24 to both 11 sides. So this is set equal to 24. 12 Now I'm going to have to complete the square, 13 uh, so that I can factor these groups, 14 uh, terms. 15 So X squared minus four X. 16 So to complete the square, I'm going to take half of the coefficient of 17 the middle term and square that so negative four divided by two squared 18 is negative two squared, which is positive four. 19 And I have to make sure to add it to both sides of the 20 equation. And this is going to be next to Y squared 21 plus six Y so six over two squared is 22 three squared, which is nine. So plus nine is equal to 24 23 plus four plus nine. 24 I'm going to raise this on the bottom just to make some room. 25 So now that I have, now that I've completed the square and I have 26 the equation, this form, I can factor. 27 So X plus or X minus actually. 28 So X minus two squared, 29 plus Y plus three squared is 30 how you factor both of these equations. 31 And that set equal to 37, which is just 24 plus four plus nine. 32 So now we have it in the form of the equation at the top, 33 and we know that 37 is equal to R squared. 34 So now all I have to do is find the square root, 35 which tells me that square root of 37 is equal to R. 36 And the correct answer is C.