If k does not equal -1, 0 or 1, does the point of intersection of line y = kx+b and line x = ky+b have a negative x-coordinate?
1. kb > 0
2. k > 1
Can we solve this question be drawing lines???
Tricky Coordinate Geometry question
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Check my two posts here:
https://www.beatthegmat.com/if-k-does-no ... 90334.html
In the second post, I solve by drawing.
https://www.beatthegmat.com/if-k-does-no ... 90334.html
In the second post, I solve by drawing.
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I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
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Mitch has beautifully explained the graphical solution, here is the Algebraic one.Mo2men wrote:If k does not equal -1, 0 or 1, does the point of intersection of line y = kx+b and line x = ky+b have a negative x-coordinate?
1. kb > 0
2. k > 1
Can we solve this question be drawing lines???
We wish to know the nature of x-coordinate.
We have two intersecting line: y = kx+b and x = ky+b;
By plugging in the value of y from the first line into the second, we get,
x = k(kx+b)+b
=> x=k^2x+bk+b
=> x(1-k^2) = b(k+1)
=> x(1+k)(1-k)=b(k+1)
=> x=b/(1-k)
Signs of b and k will determine whether x is negative/positive.
S1: kb > 0
kb > 0 implies that kb is positive. Thus, either both are positive or both are negative.
Case 1: b and k are positive.
x=b/(1-k) may or may not be negative.
(a) If 0 < k < 1, x = b / |(1-k)| => x is not negative.
(b) If k > 1, x = b / (-|(1-k)|) => x is negative. No unique answer.
There is no need to discuss case 2 (b and k are negative.) as we already concluded that S1 itself in not sufficient.
S2: k > 1
There is no information about the sign of b. Insufficient.
S1 and S2:
From S1, we know that b and k are of the same sign and from S2, we know that k is positive thus, b and k each is positive.
We saw in the Case 1 analysis that even if b and k each is positive, there is no unique answer. However, if k > 1, we have do not have Case 1.a. Thus, x is negative. Sufficient.
The correct answer: C
Relevant book: Manhattan Review GMAT Coordinate Geometry Guide
Hope this helps!
-Jay
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Mitch has beautifully explained the graphical solution, here is the Algebraic one.Mo2men wrote:If k does not equal -1, 0 or 1, does the point of intersection of line y = kx+b and line x = ky+b have a negative x-coordinate?
1. kb > 0
2. k > 1
Can we solve this question be drawing lines???
We wish to know the nature of x-coordinate.
We have two intersecting line: y = kx+b and x = ky+b;
By plugging in the value of y from the first line into the second, we get,
x = k(kx+b)+b
=> x=k^2x+bk+b
=> x(1-k^2) = b(k+1)
=> x(1+k)(1-k)=b(k+1)
=> x=b/(1-k)
Signs of b and k will determine whether x is negative/positive.
S1: kb > 0
kb > 0 implies that kb is positive. Thus, either both are positive or both are negative.
Case 1: b and k are positive.
x=b/(1-k) may or may not be negative.
(a) If 0 < k < 1, x = b / |(1-k)| => x is not negative.
(b) If k > 1, x = b / (-|(1-k)|) => x is negative. No unique answer.
There is no need to discuss case 2 (b and k are negative.) as we already concluded that S1 itself in not sufficient.
S2: k > 1
There is no information about the sign of b. Insufficient.
S1 and S2:
From S1, we know that b and k are of the same sign and from S2, we know that k is positive thus, b and k each is positive.
We saw in the Case 1 analysis that even if b and k each is positive, there is no unique answer. However, if k > 1, we have do not have Case 1.a. Thus, x is negative. Sufficient.
The correct answer: C
Relevant book: Manhattan Review GMAT Coordinate Geometry Guide
Hope this helps!
-Jay
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Thanks Jay.Jay@ManhattanReview wrote:
Case 1: b and k are positive.
x=b/(1-k) may or may not be negative.
(a) If 0 < k < 1, x = b / |(1-k)| => x is not negative.
(b) If k > 1, x = b / (-|(1-k)|) => x is negative. No unique answer.
Why did you add modulus sign for (1-k). Is absolute value?
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Hi Mo2men,Mo2men wrote:Thanks Jay.Jay@ManhattanReview wrote:
Case 1: b and k are positive.
x=b/(1-k) may or may not be negative.
(a) If 0 < k < 1, x = b / |(1-k)| => x is not negative.
(b) If k > 1, x = b / (-|(1-k)|) => x is negative. No unique answer.
Why did you add modulus sign for (1-k). Is absolute value?
A modulus is a great tool to demonstrate the outcome of a sign of an expression.
Let us bring the above.
Case 1: b and k are positive.
x=b/(1-k) may or may not be negative.
(a) If 0 < k < 1, x = b / |(1-k)| => x is not negative.
Putting (1-k) as |(1-k)| ensures that the dominator is positive. Since (1-k) is not positive for all k > 0, we have to express unambiguously. x = Positive / Positive. I did not write b as |b| since there is no ambiguity with b. It's a positive quantity. Though there is no harm in writing b as |b|.
(b) If k > 1, x = b / (-|(1-k)|) => x is negative. No unique answer.
Same here. Since for k > 1, (1-k) is negative, I fixed the negative sign with (1-k). So, within -|(1-k)| whatever is happening is already taken care. Thus, x = Positive / Negative = Negative.
Hope this is clear.
Relevant book: Manhattan Review GMAT Number Properties Guide
-Jay
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