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# In the xy-coordinate plane, line l and line k intersect at the point

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In the xy-coordinate plane, line l and line k intersect at the point [#permalink]

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12 May 2006, 05:22
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Difficulty:

85% (hard)

Question Stats:

55% (01:51) correct 45% (01:30) wrong based on 497 sessions

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In the xy-coordinate plane, line l and line k intersect at the point (4,3). Is the product of their slope negative?

(1) The product of the x-intercepts of lines l and k is positive.
(2) The product of the y-intercepts of lines l and k is negative.
Director
Joined: 16 Aug 2005
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Location: France
Re: In the xy-coordinate plane, line l and line k intersect at the point [#permalink]

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12 May 2006, 11:38
1
Well....its twice the horror for me man....below is the best I could come up with. If anyone can explain the solution clearly would be great. If this comes in the real thing, I'll most likely guess instead of trying to solve it.
------------------

Slope of line = (y2-y1)/(x2-x1)
We can assume (x2, y2) as (4,3)

From (1) either both x-intercepts are +ve or -ve so either (x1, 0) and (x2, 0)
If +ve, slope of L = (3 - 0)/(4 - x1) ... depends on value of x1
If -ve, slope of L = (3 - 0)/(4 - x1) ... which is positive.

Same goes for Line K.
Hence insufficient.

From (2) either one y-intercept is +ve and another is -ve so (0, y1) and (0, -y2)

Slope of L = (3 - y1)/(4 - 0) ... depends on value of y1
Slope of K = (3 - (-y2))/(4 - 0) ... which is positive

Hence insufficient

Taking both together, we still cannot say due to values of x1 and y1 unknown hence answer should be E.
SVP
Joined: 01 May 2006
Posts: 1788
Re: In the xy-coordinate plane, line l and line k intersect at the point [#permalink]

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12 May 2006, 15:53
1

For line L : 3 = 4*a(l) + b(l)
For line K : 3 = 4*a(k) + b(k)

1) Says us:

For L
xo(l)*a(l) + b(l) = 0
so,
xo(l)= -b(l)/ a(l)

Similarly, for k : xo(k)= -b(k)/ a(k)

xo(l) * xo(k) > 0
so, ( b(l)*b(n) ) / ( a(l)*a(n) ) > 0

Unsifficient

2) is : b(k)*b(l) < 0

Unsifficient

Combining (1) and (2), as b(k)*b(l) < 0 thus a(k)*a(l) < 0
SVP
Joined: 14 Dec 2004
Posts: 1652
Re: In the xy-coordinate plane, line l and line k intersect at the point [#permalink]

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12 May 2006, 20:01
Just thought of taking diagramatic approach for this & getting "C". Looks like following is the only possible solution.
Attachments

File comment: x-y Plane

xy.JPG [ 5.68 KiB | Viewed 9279 times ]

Intern
Joined: 01 Feb 2006
Posts: 14
Re: In the xy-coordinate plane, line l and line k intersect at the point [#permalink]

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15 May 2006, 12:41
2
2
The question becomes much simpler if you see whether the lines passing through point (4, 3) has any bearing on the sign of the slopes -- NO.

So, it comes down to the simple slope equation:
slope = y-intercept / x-intercept.

We need slope(l)*slope(k)

slope(l)*slope(k) = ((y-intercept of l) / x-intercept of l) * ((y-intercept of k / x-intercept of k)
= (y-intercept of l * y-intercept of k) / (x-intercept of l * y-intercept of x)

Using 1) and 2),
= (-ve) (+ve)
= -ve
Math Expert
Joined: 02 Sep 2009
Posts: 46128
Re: In the xy-coordinate plane, line l and line k intersect at the point [#permalink]

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07 Sep 2010, 07:22
7
4
M8 wrote:
In the xy-coordinate plane, line l and line k intersect at the point (4,3). Is the product of their slope negative?

1) The product of the x-intercepts of lines l and k is positive.
2) The product of the y-intercepts of lines l and k is negative.

I cracked this question right.
But these types of questions are real horror for me.

I'll post the OA later.

In the xy coordinate plane, line L and line K intersect at the point (4,3). Is the product of their slopes negative?

We have two lines: $$y_l=m_1x+b_1$$ and $$y_k=m_2x+b_2$$. The question: is $$m_1*m_2<0$$?

Lines intersect at the point (4,3) --> $$3=4m_1+b_1$$ and $$3=4m_2+b_2$$

(1) The product of the x-intersects of lines L and K is positive. Now, one of the lines can intersect x-axis at 0<x<4 (positive slope) and another also at 0<x<4 (positive slope), so product of slopes also will be positive BUT it's also possible one line to intersect x-axis at 0<x<4 (positive slope) and another at x>4 (negative slope) and in this case product of slopes will be negative. Two different answers, hence not sufficient.

But from this statement we can deduce the following: x-intersect is value of $$x$$ for $$y=0$$ and equals to $$x=-\frac{b}{m}$$ --> so $$(-\frac{b_1}{m_1})*(-\frac{b_2}{m_2})>0$$ --> $$\frac{b_1b_2}{m_1m_2}>0$$.

(2) The product of the y-intersects of lines L and K is negative. Now, one of the lines can intersect y-axis at 0<y<3 (positive slope) and another at y<0 (positive slope), so product of slopes will also be positive BUT it's also possible one line to intersect y-axis at y<0 (positive slope) and another at y>3 (negative slope) and in this case product of slopes will be negative. Two different answers, hence not sufficient.

But from this statement we can deduce the following: y-intercept is value of $$y$$ for $$x=0$$ and equals to $$x=b$$ --> $$b_1*b_2<0$$.

(1)+(2) $$\frac{b_1b_2}{m_1m_2}>0$$ and $$b_1*b_2<0$$. As numerator in $$\frac{b_1b_2}{m_1m_2}>0$$ is negative, then denominator $$m_1m_2$$ must also be negative. So $$m_1m_2<0$$. Sufficient.

In fact we arrived to the answer C, without using the info about the intersection point of the lines. So this info is not needed to get C.

For more on coordinate geometry check the link in my signature.
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Re: In the xy-coordinate plane, line l and line k intersect at the point [#permalink]

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07 Sep 2010, 20:23
1
C.

Sweet question. A kudos for you for that. If you draw the lines using conditions you will find that lines can intercept in quadrant 1 only when both the X intercepts are positive (both negative is not possible given the condition 2). When thats the case the angles each line make with the X-axis are acute and obtuse. Tan of such angles are opposite in sign. Hence the answer.

Thank you.
Hemanth
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Re: In the xy-coordinate plane, line l and line k intersect at the point [#permalink]

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07 Oct 2010, 17:30
1
satishreddy wrote:

159) In xy coordinate plane, line l and line k intersect at point (4, 3). Is product of the slopes negative? C
a. Product of x intercepts of lines l and k is positive
b. Product of y intercepts of lines l and k is negative

OA: C

Question: Product of the slope of line l and k -ve.

Let line l be y1= m1x1+c1 and k be y2 = m2x2+c2. General equation of the line => y = mx+c.

Hence to determine whether m1*m2 = -ve

Now x intercept is when y=0 and y intercept is when x=0.

Statement A: Product of x intercepts of line l and k is +ve.

x-intercept of line l, that is when y=0=> x1=-c1/m1 and x-intercept of line k is x2=-c2/m2.

Given their product is +ve. Now -c1/m1 * -c2/m2 is +ve. Either c1*c2 and m1*m2 could be both +ve or both -ve. We cannot determine the whether m1*m2 is +ve or -ve since we do not have information about c1*c2.

Statement B: Product of y intercepts of line l and k is -ve.

y-intercept of line l, that is when x=0=> y1=c1 and y-intercept of line k is y2=c2.

Given their product is -ve that is c1*c2=-ve. However with this statement alone we cannot determine whether m1*m2=-ve or +ve.

Combining both the statement we know that since c1*c2 = -ve, m1*m2 has to be -ve to satisfy Statement A.

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Manager
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Re: In the xy-coordinate plane, line l and line k intersect at the point [#permalink]

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08 Oct 2010, 08:05
1
Equation for a line in intercept form, X/A+Y/B=1 and slope will be m=-(B/A)

Now take the equation of first line as X/A1+Y/B1=1 and second line as X/A2+Y/B2=1

Product of their slope will be (B1/A1)*(B2/A2)=(B1*B2)/(A1*A2)

with first (A1 *A2) positive
With 2nd (B1*B2) -ve

So with both we can say Product of their slope is -ve. that is C.

Consider KUDOS if you like my solution.
Intern
Joined: 14 Sep 2010
Posts: 19
Re: In the xy-coordinate plane, line l and line k intersect at the point [#permalink]

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22 Dec 2011, 07:13
Line L and Line K intersect at the point (4,3). Is the product of their slopes negative?

(1) The product of the x-intercepts of Line L and Line K is positive.

Combination A1 m b

(-2, 0) (4,3) y = 1/2x + 1
(-3, 0) (4,3) y = 3/7x + 9/7

Combination B1 m b

(1, 0) (4,3) y = x - 1
(6, 0) (4,3) y = -3/2x + 9

(2) The product of the y-intercepts of Line L and Line K is negative.

Combination A2 m b

(0, 6) (4,3) y = -3/4x + 6 (8,0)
(0, -2) (4,3) y = 5/4x - 2 (8/5, 0)

Combination B2 m b

(0,-2) (4,3) y = 5/4x - 2 (8/5,0)
(0, 2) (4,3) y = 1/4x + 2 (-8, 0)

(C) Log: From S(2) result, select only neg.
y-intercept (where b1*b2<0,
Log: Combination A2 and B2
Log: From A2 and B2, select only pos.
x-intercept (b1*b2)/(m1*m2) > 0
Log: A2

This restriction is made more cogent by noting that a line with a negative y-intercept and destination (4,3) must cross the x-axis in positive territory and have a positive slope.

Since the other line must have a positive y-intercept and is restricted to a positive x-intercept, its slope
must be negative.

Both statements combined are sufficient.

Algebraic Solution

Is m1*m2 < 0

(1) -(b1/m1) * -(b2/m2) > 0

(b1*b2)/(m1*m2) > 0

(2) b1*b2 < 0

(1)&(2)

b1*b2/m1*m2 > 0, and b1*b2 < 0

m1*m2 < 0
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Re: In the xy-coordinate plane, line l and line k intersect at the point [#permalink]

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10 Jun 2015, 02:49
1
very hard question on gmatprep.

this question can be done by drawing the lines and see that C is correct.
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Re: In the xy-coordinate plane, line l and line k intersect at the point [#permalink]

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23 Jun 2015, 07:02
thangvietnam wrote:
very hard question on gmatprep.

this question can be done by drawing the lines and see that C is correct.

Yes, solving by drawing is much easier in this question
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Re: In the xy-coordinate plane, line l and line k intersect at the point [#permalink]

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13 Nov 2015, 01:46
Bunuel wrote:
M8 wrote:
In the xy-coordinate plane, line l and line k intersect at the point (4,3). Is the product of their slope negative?

1) The product of the x-intercepts of lines l and k is positive.
2) The product of the y-intercepts of lines l and k is negative.

I cracked this question right.
But these types of questions are real horror for me.

I'll post the OA later.

In the xy coordinate plane, line L and line K intersect at the point (4,3). Is the product of their slopes negative?

We have two lines: $$y_l=m_1x+b_1$$ and $$y_k=m_2x+b_2$$. The question: is $$m_1*m_2<0$$?

Lines intersect at the point (4,3) --> $$3=4m_1+b_1$$ and $$3=4m_2+b_2$$

(1) The product of the x-intersects of lines L and K is positive. Now, one of the lines can intersect x-axis at 0<x<4 (positive slope) and another also at 0<x<4 (positive slope), so product of slopes also will be positive BUT it's also possible one line to intersect x-axis at 0<x<4 (positive slope) and another at x>4 (negative slope) and in this case product of slopes will be negative. Two different answers, hence not sufficient.

But from this statement we can deduce the following: x-intersect is value of $$x$$ for $$y=0$$ and equals to $$x=-\frac{b}{m}$$ --> so $$(-\frac{b_1}{m_1})*(-\frac{b_2}{m_2})>0$$ --> $$\frac{b_1b_2}{m_1m_2}>0$$.

(2) The product of the y-intersects of lines L and K is negative. Now, one of the lines can intersect y-axis at 0<y<3 (positive slope) and another at y<0 (positive slope), so product of slopes will also be positive BUT it's also possible one line to intersect y-axis at y<0 (positive slope) and another at y>3 (negative slope) and in this case product of slopes will be negative. Two different answers, hence not sufficient.

But from this statement we can deduce the following: y-intercept is value of $$y$$ for $$x=0$$ and equals to $$x=b$$ --> $$b_1*b_2<0$$.

(1)+(2) $$\frac{b_1b_2}{m_1m_2}>0$$ and $$b_1*b_2<0$$. As numerator in $$\frac{b_1b_2}{m_1m_2}>0$$ is negative, then denominator $$m_1m_2$$ must also be negative. So $$m_1m_2<0$$. Sufficient.

In fact we arrived to the answer C, without using the info about the intersection point of the lines. So this info is not needed to get C.

For more on coordinate geometry check the link in my signature.

In the above explanation
'But from this statement we can deduce the following: y-intercept is value of $$y$$ for $$x=0$$ and equals to $$x=b$$ ' I don't understand how x=b. I think it would be y = b. Am I right?
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Re: In the xy-coordinate plane, line l and line k intersect at the point [#permalink]

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16 Oct 2017, 02:44
so, we do not need the information that the two lines intersect at (4,3)

is that right?

the question is not good. right.

redundant information?
Re: In the xy-coordinate plane, line l and line k intersect at the point   [#permalink] 16 Oct 2017, 02:44
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