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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