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28 Feb 2019, 07:25
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Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
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Question Stats:
72% (02:04) correct
28%
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based on 116
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(x-a)^2 + (y-b)^2 = r^2 is the equation of a circle with radius r and center (a,b). What is the radius of the circle with equation x^2 + y^2 – 10x + 4y = 20?
A. 4
B. 5
C. 6
D. 7
E. 8
A. 4
B. 5
C. 6
D. 7
E. 8
28 Feb 2019, 14:45
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The secret to solving such questions is to make the given equation in question similar to the definition of circle.
The second key is to see the terms with single power (10x and 4y).
Now lets solve......
Make the equation as...
x^2 -(2 * x *5) + 5^2 - 5^2 + y^2 + (2 * y *2 ) + 2^2 -2^2 = 20
which means
(x-5)^2 + (y+2)^2 = 20+ 5^2+2^2 = 49 = 7^2
Thus 7 is the radius....
Give Kudos if you like the solution...Cheers !!! Happy Learning !!!
The second key is to see the terms with single power (10x and 4y).
Now lets solve......
Make the equation as...
x^2 -(2 * x *5) + 5^2 - 5^2 + y^2 + (2 * y *2 ) + 2^2 -2^2 = 20
which means
(x-5)^2 + (y+2)^2 = 20+ 5^2+2^2 = 49 = 7^2
Thus 7 is the radius....
Give Kudos if you like the solution...Cheers !!! Happy Learning !!!
General Discussion
28 Feb 2019, 18:37
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MathRevolution
This is an interesting problem, because while I've seen 'completing the square' used in solutions for GMAT problems before, I'm not sure that I've seen it required in an official problem. I'd be curious to see if anybody can show an official problem that requires you to do this.
Here's how it works: you need to take the equation you have, and put it in a form so that all of the variables are contained in the \((x-a)^2\) and the \((y-b)^2\) parts of the equation, and you only have numbers left over.
Focus on the x's first. The terms that have an x in them are \(x^2 - 10x\). You want to make that look like (x-something)^2.
Well, when you square (x-a), you end up with \(x-2ax+a^2\). So, 2a has to equal 10 in order for the expressions to look alike. That means we're looking at \((x-5)^2\).
\((x-5)^2\) equals\(x^2 - 10x + 25\). Therefore, \(x^2 - 10x\) can be rewritten as \((x-5)^2-25\).
Similarly, for the y terms, we'll rewrite as \((y+2)^2\), which equals \(y^2 + 4y + 4\). Therefore, \(y^2+4y\) can be rewritten as\((y+2)^2-4\).
Now, put those back into the original equation.
\(x^2 + y^2 – 10x + 4y = 20\)
\((x-5)^2 - 25 + (y+2)^2 - 4 = 20\)
\((x-5)^2 + (y+2)^2 = 20 + 4 + 25 = 49 = 7^2\)
So, the radius is 7.
