The line k has a positive slope m and it passes through the point (-4,

Question

The line k has a positive slope m and it passes through the point (-4,6). the area of the triangle formed by k, the x axis and y axis is 54 square units. What is the possible value for m?

A. 1/4
B. 3/4
C. 5/4
D. 7/4
E. cannot be determined

BrushMyQuant · Accepted Answer

(Figure attached)
Equation of line will be given by
y-6 / (x- (-4)) = m
or, y = mx + 4m + 6

Area of triangle formed by line k and x and y-axis = area of Triangle AOB = (1/2) AO * OB = (1/2)*(4m+6)*(4m+6)/m = 54 (given)

Simplifying we will get
16m^2 + 48m + 36 = 108m
=> 16m^2 - 60m + 36 = 0
=> 4m^2 - 15m + 9 = 0
=> 4m^2 - 12m - 3m + 9 =0
=> 4m ( m-3) -3 (m-3) = 0
=> m = 3 , 3/4

So, answer will be B
Hope it helps!

AVRonaldo · Answer

I am not sure if my way of approach is correct. But still I ended up with the answer..

Area of the triangle formed by the line k with X & Y axis is 54.
1/2 * Base * Height = 54

Here Base refers to the x-intercept in units and Height refers to the y-intercept in units
So --> X (or Base) * Y (or Height) = 108

We know that the line k passes through the point (-4,6) and has a positive slope. 
Therefore, the point where the line meets the X-axis must be definitely greater than -4 i.e greater than 4 units from the origin(Base)
Similarly, the point where the line meets the Y-axis must be definitely greater than 6 i.e greater than 6 units from the origin(height)

There are a handful of factors that multiply into 108 but only one satisfies the above condition. 
9 and 12 -- valid since it satisfies both of the above criteria

the below ones do not 
27 and 4 
18 and 6
36 and 3
108 and 1

So the y intercept can be either (0,12) or (0,9) 
and x-intercept can be either (-9,0) or (-12,0)

We slope m = (y2-y1)/(x2-x1) --> 12/9 or 9/12 i.e 4/3 or 3/4

Answer is 3/4 (B)

GMATinsight · Answer

Your reasoning is ABSOLUTELY SOUND.

Now what surprises me is how can someone which such reasoning skills end up with a score of 39 in Quant.

Lucky2783 · Answer

check out the attached solution . let me know if case of questions. 
thanks 
Lucky
Consider +1 Kudos if you like this post.

hatemnag · Answer

Hi Bunuel

Please recognize the approach here. Is it right to try the different factors that produce 108 or you have more reliable and shorter way.
Actually, I solved in such way but it took me 4 Mins to complete the operation.

SandhyAvinash · Answer

someone please help me to understand how did you find 108?? please post your replies with formula and additional steps to make it clear for freshers.

gracie · Answer

The line k has a positive slope m and it passes through the point (-4,6). the area of the triangle formed by k, the x axis and y axis is 54 square units. What is the possible value for m?

A. 1/4
B. 3/4
C. 5/4
D. 7/4
E. cannot be determined

assume the line rising from the x axis to (-4,6) is the height of a 6-8-10 right triangle
thus, the triangle base on the x axis will run 8 units from (-4,0) to (-12,0)
we now know two points on line k: (-4,6) and (-12,0)
m= =6/8=3/4
B

kelindoan · Answer

The triangle area is (x*y)/2=54 => x*y=108

Now rise/run=slope 
=> (y-6)/(x+4)=m
=> y=xm+4m+6

=> (4m+6)*(4m+6)/m=108
Solve this we find m = 3 or 3/4

Thus answer is B

SandhyAvinash · Answer

hi kelindoan step 3 is still unclear how did you find this statement to solve for m??? please help

Bharath99 · Answer

Hi, Hope you understood till equation y=xm+4m+6 Now we need values of x and y intercepts here (nothing but value of x and y) x intercept => 0= xm+4m+6 (place y=0) from her we get x = -(4m+6) / m similarly for y intercept , place x = 0 .... so y =4m+6 But the distances x and y are absolute values.. so we can remove negative sign Hope you understood now.. Please let know in case of any doubts.. If it helps, kudos please

Shruti0805 · Answer

Hi  Bunuel ,  BrushMyQuant ,
Using the 2 points at the X and Y axes and the point through which the line passes, following are the equations for slope that I came up with:

\frac{(y-6)}{(0+4)} = \frac{(6-0)}{(-4-x)}

from this, the equation I get is  6x-4y+xy = 0 
=>  4y=6x -108

that gives us a slope of 3/2 which isnt an answe choice. Could you please help understand where am I going wrong?

BrushMyQuant · Answer

Hi  Shruti0805 ,

Couple of things:

1. You are assuming that if a line is passing through point (-4,6) then it will pass through (0,6) and (-4,0) which is not true. See attached figure for reference.
2. The equation of a line passing through two points is (y-y1)/(x-x1) = y2-y1)/(x2-x1) so it will be (y-6)/(x-0) = (0-6)/(-4-0) and not (y-6)/(0+4) = (6-0)/(-4-x)
3. Equation of a line will always be a one degree equation in terms of x and y like ax + by+ c = 0, where a,b,c are constants.

hope it helps!

Kinshook · Answer

Given: The line k has a positive slope m and it passes through the point (-4,6). the area of the triangle formed by k, the x axis and y axis is 54 square units.

Asked: What is the possible value for m?

Equation of line k
y-6 = m (x+4) 
Where m>0

x-intercept of line k: 
-6 = mx + 4m 
x = (-6-4m)/m = -6/m -4

y-intercept of line k : 
y-6 = 4m
y = 6+4m

Area of triangle formed by x-axis, y-axis and line k 
= 1/2 * |x-intercept * y-intercept|
= 1/2 |(-6/m -4)(6+4m)| = 54
= |(6+4m)(6+4m)/m| = 108
= (6+4m)^2 = 108m
(3+2m)^2 = 27m
9 + 4m^2 + 12m = 27m
4m^2 - 15m + 9 = 0
4m^2 -12m -3m + 9 = 0
4m (m-3) - 3(m-3) = 0
(4m-3)(m-3) = 0
m = 3/4 or 
m = 3

IMO B

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The line k has a positive slope m and it passes through the point (-4,

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