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sandra123
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If x and y are numbers such that (x+9)(y-9)=0, what is the Updated on: 23 Oct 2024, 08:06
Originally posted by sandra123 on 15 Sep 2013, 12:35.
Last edited by Bunuel on 23 Oct 2024, 08:06, edited 2 times in total.
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67% (01:17) correct 33% (01:32) wrong based on 347 sessions

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If x and y are numbers such that (x + 9)(y - 9) = 0, what is the smallest possible value of x^2 + y^2

A. 0
B. 9
C. 18
D. 81
E. 162

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I understand its a basic question, but I fail to understand the concept. I simply put x+9=0, so x=-9. in the same way, y-9=0, so y=9
so x^2 + y^2 = 81+81=162
what am I doing wrong?
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D
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Bunuel
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If x and y are numbers such that (x+9)(y-9)=0, what is the 15 Sep 2013, 12:41
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sandra123
If x and y are numbers such that (x+9)(y-9)=0, what is the smallest possible value of x^2 + y^2

A. 0
B. 9
C. 18
D. 81
E. 162

I understand its a basic question, but I fail to understand the concept. I simply put x+9=0, so x=-9. in the same way, y-9=0, so y=9
so x^2 + y^2 = 81+81=162
what am I doing wrong?
Show more

From (x + 9)(y - 9) = 0, it follows that either x = -9 or y = 9 (or both). Thus, either x^2 = 81 or y^2 = 81.

Now, if x^2 = 81, then the least value of y^2 is 0, making the least value of x^2 + y^2 = 81 + 0 = 81.

Similarly, if y^2=81, then the least value of x^2 is 0, so the least value of x^2 + y^2 = 0 + 81 = 81.

Answer: D.

Hope it's clear.
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tchang
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If x and y are numbers such that (x+9)(y-9)=0, what is the 15 Sep 2013, 17:49
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In the given equation, if x = -9 then y could = ANYTHING

If y = 9, then x could = ANYTHING

They DON'T BOTH have to be true at the same time.

9^2 + 0^2 = 81

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If x and y are numbers such that (x+9)(y-9)=0, what is the 22 Sep 2013, 05:26
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sandra123
If x and y are numbers such that (x+9)(y-9)=0, what is the smallest possible value of x^2 + y^2

A. 0
B. 9
C. 18
D. 81
E. 162

I understand its a basic question, but I fail to understand the concept. I simply put x+9=0, so x=-9. in the same way, y-9=0, so y=9
so x^2 + y^2 = 81+81=162
what am I doing wrong?
Show more

This question could me solved in this way.

(x+9)(x-9) = 0
=> xy - 81 = 9(x-y)
=> (x-y) = (xy/9) - 9
=> Squaring on both sides and cancelling equal terms
=> x^2 + y^2 = (xy)^2/81 + 81

Notice that 81 is added to (xy)^2/81, so the absolute minimum value of the above expression will be equal to 81 when x=9 and y=9 => x^2 + y^2 = 162.
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If x and y are numbers such that (x+9)(y-9)=0, what is the 04 Sep 2017, 08:54
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Since (x+9)(y-9) = 0 => x= -9 OR y=9

If x= -9 => X^2 = 81
The smallest value y can now take is y =0
Hence x^2+y^2 = 81

Same is the case with Y=9
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If x and y are numbers such that (x+9)(y-9)=0, what is the 02 Nov 2024, 10:06
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Hi Bunuel,

I just couldn't understand, why couldn't we just put the value of x and y obtained from given equation and put it like (-9^2)+(9^2)= 81+81=162?

Please help me understand.


sandra123
If x and y are numbers such that (x + 9)(y - 9) = 0, what is the smallest possible value of x^2 + y^2

A. 0
B. 9
C. 18
D. 81
E. 162

Show Spoiler

I understand its a basic question, but I fail to understand the concept. I simply put x+9=0, so x=-9. in the same way, y-9=0, so y=9
so x^2 + y^2 = 81+81=162
what am I doing wrong?
Show more
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Bunuel
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If x and y are numbers such that (x+9)(y-9)=0, what is the 02 Nov 2024, 10:25
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7Amulya
Hi Bunuel,

I just couldn't understand, why couldn't we just put the value of x and y obtained from given equation and put it like (-9^2)+(9^2)= 81+81=162?

Please help me understand.
Bunuel
sandra123
If x and y are numbers such that (x+9)(y-9)=0, what is the smallest possible value of x^2 + y^2

A. 0
B. 9
C. 18
D. 81
E. 162

I understand its a basic question, but I fail to understand the concept. I simply put x+9=0, so x=-9. in the same way, y-9=0, so y=9
so x^2 + y^2 = 81+81=162
what am I doing wrong?

From (x+9)(y-9)=0, it follows that either x=-9 or y=9 (or both). Thus, either x^2=81 or y^2=81.

Now, if x^2 = 81, then the least value of y^2 is 0, making the least value of x^2 + y^2 = 81 + 0 = 81.

Similarly, if y^2=81, then the least value of x^2 is 0, so the least value of x^2 + y^2 = 0 + 81 = 81.

Answer: D.

Hope it's clear.
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Because from (x + 9)(y - 9) = 0, if x = -9, it’s not necessary for y to be 9; in this case, y can take any value. Similarly, if y = 9, it’s not necessary for x to be -9; in this case, x can take any value. Therefore, it’s not necessary for x = -9 and y = 9 to be true simultaneously; only one must be true. So, we should set either x = -9 or y = 9 and then set the other variable to get the smallest possible value of x^2 + y^2. This happens when x = -9 and y = 0, or when x = 0 and y = 9.

Hope it's clear.
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