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If the lines given by $3 x + 2 ky = 2$ and $2 x + 5 y + 1 = 0$ are parallel, then what is the value of $k$ ?

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− 54

25

154

32

answer is C.

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Detailed Solution

Given equations are

3 x + 2 ky = 2

and

2 x + 5 y + 1 = 0

are parallel.
A pair of linear equations of two variables x and y is represented as the general form are,

a 1 x + b 1 y + c 1 = 0

and

a 2 x + b 2 y + c 2 = 0

.
We know that in a pair of linear equations in two variables, if

a 1 a 2 = b 1 b 2 ≠ c 1 c 2

, then the lines are parallel and have no solution.
On comparing the given equations with the general form, we get,

a 1 = 3

b 1 = 2 k

c 1 = − 2

a 2 = 2

b 2 = 5

, and

c 2 = 1

The ratios of the coefficients are,

a 1 a 2 = 32

b 1 b 2 = 2 k 5

and

\frac{c_{1}}{c_{2}} = - \frac{2}{1}

⇒ 32 = 2 k 5

On solving for k, we get,

k = 154

Hence, the correct option is 3.

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