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Updated on: 05 May 2023, 12:14
Originally posted by arjtryarjtry on 28 Aug 2008, 09:27.
Last edited by Bunuel on 05 May 2023, 12:14, edited 5 times in total.
Last edited by Bunuel on 05 May 2023, 12:14, edited 5 times in total.
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A
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Question Stats:
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66%
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In the coordinate plane, is the point \((0, 0)\) closer to the point \((u, v)\) than to the point \((u, v + 1)\)?
(1) \(v + u^2 = -1\)
(2) \(v < 0\)
M22-11
(1) \(v + u^2 = -1\)
(2) \(v < 0\)
M22-11
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14 May 2014, 00:51
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In the coordinate plane, is the point \((0, 0)\) closer to the point \((u, v)\) than to the point \((u, v + 1)\)?
The formula to calculate the distance between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}\).
Essentially, the question asks whether the distance between the points \((0, 0)\) and \((u, v)\) is less than the distance between the points \((0, 0)\) and \((u, v + 1)\):
Is \(\sqrt{(u-0)^2+(v-0)^2} \lt \sqrt{(u-0)^2+(v+1-0)^2}\)?
Is \(\sqrt{u^2+v^2} \lt \sqrt{u^2+(v+1)^2}\)?
Is \(u^2+v^2 \lt u^2+v^2+2v+1\)?
Is \(v \gt -\frac{1}{2}\)?
(1) \(v + u^2 = -1\).
Rearrange: \(v=-1-u^2\). Now, since \(u^2 \ge 0 \), then \(v=-1-u^2 \le -1\), so the answer to the question is NO. Sufficient.
(2) \(v < 0\). Not sufficient.
Answer: A
The formula to calculate the distance between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}\).
Essentially, the question asks whether the distance between the points \((0, 0)\) and \((u, v)\) is less than the distance between the points \((0, 0)\) and \((u, v + 1)\):
Is \(\sqrt{(u-0)^2+(v-0)^2} \lt \sqrt{(u-0)^2+(v+1-0)^2}\)?
Is \(\sqrt{u^2+v^2} \lt \sqrt{u^2+(v+1)^2}\)?
Is \(u^2+v^2 \lt u^2+v^2+2v+1\)?
Is \(v \gt -\frac{1}{2}\)?
(1) \(v + u^2 = -1\).
Rearrange: \(v=-1-u^2\). Now, since \(u^2 \ge 0 \), then \(v=-1-u^2 \le -1\), so the answer to the question is NO. Sufficient.
(2) \(v < 0\). Not sufficient.
Answer: A
29 Aug 2008, 01:44
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Let's consider each statement carefully:
a) On the coordinate plane is point (0, 0) closer to point (u, v) than to point (u, v + 1) ?
First of all, u does not influence on answer. Therefore, we can restate: On the coordinate line is point (0) closer to point (v) than to point (v + 1) ?
Now, we can translate it to language of formulas: |v|<|v+1|
Eventually, we can write: v>-0.5
b) v + u^2 = -1 --> v=-1-u^2 --> v<-1
Now, we can restate our problem as following:
Does v>-0.5 ?
1. v<-1
2. v<0
Answer is obviously A.
a) On the coordinate plane is point (0, 0) closer to point (u, v) than to point (u, v + 1) ?
First of all, u does not influence on answer. Therefore, we can restate: On the coordinate line is point (0) closer to point (v) than to point (v + 1) ?
Now, we can translate it to language of formulas: |v|<|v+1|
Eventually, we can write: v>-0.5
b) v + u^2 = -1 --> v=-1-u^2 --> v<-1
Now, we can restate our problem as following:
Does v>-0.5 ?
1. v<-1
2. v<0
Answer is obviously A.
