What is the value of k for which the pair of linear equation 4x 6y 1 0 and 2x Ky 5 0 represents parallel lines is?

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10 Questions 40 Marks 10 Mins

Concept:

Two line equations are parallel when their tangent is equal to eachother.

Let,

y = m1x + c   ------ (1)

y = m2x + c   ------ (2)

If these lines are parallel. Then,

m1 = m2.

Given:

4x + 6y - 1 = 0 and 2x + ky - 7 = 0

Calculation:

After arranging the given equation (1) & (2)

y = -\(\frac{4}{6}\)x + \(\frac{1}{6}\)   ------ (3)

y = -\(\frac{2}{k}\)x + \(\frac{7}{k}\)   ------ (4) 

After comparing with standard equation,

m1 = -\(\frac{4}{6}\)  And m2 = -\(\frac{2}{k}\)

If these lines are parallel. Then,

m1 = m2

⇒ -\(\frac{4}{6}\) = - \(\frac{2}{k}\) 

⇒ 4k = 12

⇒ k = 3

Two line equations are perpendicular when the product of their tangent is  (-1).

Let,

y = m1x + c   ------ (1)

y = m2x + c   ------ (2)

If these lines are parallel. Then,

m1 × m2 = -1

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The value of k for which the pair of linear equations 4x + 6y – 1 = 0 and 2x + ky – 7 = 0 represents parallel lines is

k = 3

Explanation;

Hint:

Slope of 4x + 6y – 1 = 0

6y = – 4x + 1

⇒ y = `(-4)/6x + 1/6`

Slope = `(-4)/6 = (-2)/3`

Slope of 2x + ky – 7 = 0

ky = – 2x + 7

y = `(-2)/"k"x + 7/"k"`

Slope of a line = `(-2)/"k"`

Since the lines are parallel

`(-2)/3 = (-2)/"k"`

– 2k = – 6

k = `6/2`

= 3

Concept: Simultaneous Linear Equations

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