vjsharma25 wrote:
How many integral values of k are possible, if the lines 3x+4ky+6 = 0, and kx-3y+9 = 0 intersect in the 2nd quadrant.
A. 5
B. 4
C. 3
D. 6
E. 2
The solution provided by
beyondgmatscore is perfect. A student asked me to add a few steps, so that what I'll do.
GIVEN:
3x + 4ky + 6 = 0
kx - 3y + 9 = 0
Rewrite as:
3x + 4ky = -6
kx - 3y = -9
NOTE: The intersection of the two lines will be at the point where the same pair of x- and y-values satisfy BOTH equations.
In other words, we want to SOLVE the system of equations above.
So, let's use the ELIMINATION method to solve the system.
First we'll eliminate the x terms
So, multiply both sides of the TOP equation by k, and multiply both sides of the BOTTOM equation by 3 to get:
3kx + 4k²y = -6k
3kx - 9y = -27
Subtract the bottom equation from the top equation to get: 4k²y + 9y = -6k + 27
Rewrite as: y(4k² + 9) = 27 - 6k
Divide both sides by (4k² + 9) to get:
y = (27 - 6k)/(4k² + 9)KEY CONCEPT: If the two lines intersect in the 2nd quadrant, then the x-value must be NEGATIVE, and the y-value must be POSITIVE
If the y-value must be POSITIVE, we can write:
(27 - 6k)/(4k² + 9) > 0Since (4k² + 9) is POSITIVE for all values of k, we can conclude that 27 - 6k is POSITIVE
In other words: 27 - 6k > 0
Add 6k to both sides to get: 27 > 6k
Divide both sides by 6 to get:
4.5 > k--------------------------------------
Now we'll perform similar steps to isolate x (by eliminating the y terms)
Take:
3x + 4ky = -6
kx - 3y = -9
Multiply both sides of the TOP equation by 3, and multiply both sides of the BOTTOM equation by 4k to get:
9x + 12ky = -18
4k²x - 12ky = -36k
Add the two equations to get: 9x + 4k²x = -18 - 36k
Factor left side: x(9 + 4k²) = -18 - 36k
Divide both sides by (9 + 4k²) to get:
x = (-18 - 36k)/(9 + 4k²)KEY CONCEPT: If the two lines intersect in the 2nd quadrant, then the x-value must be NEGATIVE, and the y-value must be POSITIVE
If the x-value must be NEGATIVE, we can write:
(-18 - 36k)/(9 + 4k²) < 0Since (9 + 4k²) is POSITIVE for all values of k, we can conclude that -18 - 36k is NEGATIVE
In other words: -18 - 36k < 0
Add 18 to both sides to get: -36k < 18
Divide both sides by -36 to get:
k > -0.5--------------------------------------
Now combine both results to get:
-0.5 < k < 45How many integral values of k are possible?If
-0.5 < k < 45, then the possible INTEGER values of k are: 0, 1, 2, 3, 4 (5 possible values)
Answer: A
Cheers,
Brent
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