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If x2 - y2 = 0 and xy 0, which of the following MUST be true
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Updated on: 30 Nov 2024, 02:33
Updated on: 30 Nov 2024, 02:33
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If \(x^2 - y^2 = 0\) and \(xy \neq 0\), which of the following MUST be true?
Indicate all such statements.
A. \(x = y\)
B. \(|x| = |y|\)
C. \(\frac{x^2}{y^2}=1\)
Indicate all such statements.
A. \(x = y\)
B. \(|x| = |y|\)
C. \(\frac{x^2}{y^2}=1\)
Re: If x2 - y2 = 0 and xy 0, which of the following MUST be true
[#permalink]
23 May 2018, 13:40
23 May 2018, 13:40
2
The answers are B and C.
Let's go through each answer one by one.
A. x=y
At first glance, this answer looks correct for the equation \(x^2 - y^2=0\). However, we have to keep in mind that any number, positive or negative, when squared, is positive. Because the only restriction is that \(xy > 0\), this means that either x or y could be positive while the other is a negative.
For example,
When x and y are both 5, then \(5^2 - 5^2 = 0\). However, if x=-5 and y=5, \((-5)^2 - 5^2 = 0\)
So A is false.
B. |x| = |y|
Using the last example, we can see that this is true. |-5| = |5|
The same goes for any combination of numbers.
B is true.
C. \(\frac{x^2}{y^2}=0\)
Following up from A, we already know that any number, positive or negative squared stays positive. If both numbers when squared, subtracted from each other are zero, then one number over the other will always be 1.
Example.
\(\frac{-5^2}{5^2}\) = 0
and \(\frac{5^2}{5^2}\) = 0
C is true.
Let's go through each answer one by one.
A. x=y
At first glance, this answer looks correct for the equation \(x^2 - y^2=0\). However, we have to keep in mind that any number, positive or negative, when squared, is positive. Because the only restriction is that \(xy > 0\), this means that either x or y could be positive while the other is a negative.
For example,
When x and y are both 5, then \(5^2 - 5^2 = 0\). However, if x=-5 and y=5, \((-5)^2 - 5^2 = 0\)
So A is false.
B. |x| = |y|
Using the last example, we can see that this is true. |-5| = |5|
The same goes for any combination of numbers.
B is true.
C. \(\frac{x^2}{y^2}=0\)
Following up from A, we already know that any number, positive or negative squared stays positive. If both numbers when squared, subtracted from each other are zero, then one number over the other will always be 1.
Example.
\(\frac{-5^2}{5^2}\) = 0
and \(\frac{5^2}{5^2}\) = 0
C is true.
Re: If x2 - y2 = 0 and xy 0, which of the following MUST be true
[#permalink]
26 May 2018, 12:59
26 May 2018, 12:59
Good one!
