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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.

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

Guys please elaborate your solution for me. Thank you.

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.

Answer: C.

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.
_________________

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)

Bunuel i just dont understand "But from this statement we can deduce the following: x-intersect is value of for " this part rest of them are ok. Can you tell me how did you decide y=0?

X-intercept is the point where a line (a graph) crosses the x-axis. So it's the point on x-axis, any point on x-axis has y-coordinate equal to zero, which means that X-intercept is the point \((x,0)\) - the value of \(x\) when \(y=0\): \(y=mx+b\) --> \(0=mx+b\) --> \(x=-\frac{b}{m}\). So X-intercept of a line \(y=mx+b\) is \(x=-\frac{b}{m}\);

Y-intercept is the point where a line (a graph) crosses the y-axis. So it's the point on y-axis, any point on y-axis has x-coordinate equal to zero, which means that Y-intercept is the point \((0,y)\) - the value of \(y\) when \(x=0\): \(y=mx+b\) --> \(y=m*0+b\) --> \(y=b\). So Y-intercept of a line \(y=mx+b\) is \(y=b\).

Check Coordinate Geometry chapter for more (link in my signature).

Re: math DS questions: need help solving [#permalink]

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07 Oct 2010, 17:30

1

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satishreddy wrote:

hi guys, below are few DS questions i had trouble answering,,,,,,,,,,need help.....thanks in advance

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.

Hence answer is C.
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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.

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.

Guys please elaborate your solution for me. Thank you.

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.

Answer: C.

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.

Bunuel i just dont understand "But from this statement we can deduce the following: x-intersect is value of for " this part rest of them are ok. Can you tell me how did you decide y=0?
_________________

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.

Re: In the xy-coordinate plane, line l and line k intersect at [#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.

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

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29 Nov 2013, 22:41

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

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02 Dec 2014, 06:40

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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.

.

Question: Is the product of slopes of line L and K negative?

Requirement: For the product of the slopes to be negativeone of the lines must have positive slope (Sloping upward from left to right) and one of the lines must have Negative slope (Sloping Downward from left to right)

Statement 1: The product of the x-intercepts of lines l and k is positive.

With positive X intercept of both the lines, both the lines may have positive slope OR With positive X intercept of both the lines, One of the lines may have positive slope and other line may have negative slope therefore NOT SUFFICIENT

Statement 2: The product of the y-intercepts of lines l and k is negative.

With positive Y intercept of one of the lines and Negative Y intercept of other line, both the lines may have positive slope OR With positive Y intercept of one of the lines and Negative Y intercept of other line, One of the lines may have positive slope and other line may have negative slope therefore NOT SUFFICIENT

Combining the two Statements:

Case 1: Both Lines have positive X intercepts (For product of therefore X intecepts to be (+ve) ) One of the lines will have positive Y intercept and one Line will have Negative Y intercept

The line with (+ve) X-intecept and (+ve)Y intercept will bound to have (-ve)Slope and The line with (+ve) X-intecept and (-ve)Y intercept will bound to have (+ve)Slope Hence Product of slopes will be Negative

Case 2: Both Lines have Negative X intercepts (For product of therefore X intecepts to be (+ve) ) One of the lines will have positive Y intercept and one Line will have Negative Y intercept

The line with (-ve) X-intecept and (+ve) Y intercept will bound to have (+ve)Slope and The line with (-ve) X-intecept and (-ve) Y intercept will bound to have (-ve)Slope Hence Product of slopes will be Negative

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Re: In the xy-coordinate plane, line l and line k intersect at [#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.

Guys please elaborate your solution for me. Thank you.

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.

Answer: C.

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?
_________________

Hasan Mahmud

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