The slope of a line is double of the slope of another line. If tangent of the angle between them is $$\dfrac{1}{3}$$ , find the slopes of the lines.

Asked by Pragya Singh | 1 year ago |  88

##### Solution :-

Let us consider ‘m1’ and ‘m’ be the slope of the two given lines such that m= 2m

We know that if θ is the angle between the lines l1 and l2 with slope m1 and m2, then

$$|\dfrac{m_2+m_1}{1+m_1m_2}|$$

It is also given that the tangent of the angle between the two lines is$$\dfrac{1}{3}$$

$$\dfrac{1}{3}=|\dfrac{m-2m}{1+(2m)\times m}|$$

$$\dfrac{1}{3}=|\dfrac{-m}{1+2m^2}|$$ or

$$|\dfrac{-m}{1+2m^2}|$$

Case 1:-

$$\dfrac{1}{3}=|\dfrac{-m}{1+2m^2}|$$

1+2m2 = -3m

2m2 +1 +3m = 0

2m (m+1) + 1(m+1) = 0

(2m+1) (m+1)= 0

m = -1 or $$- \dfrac{1}{2}$$

If m = -1, then the slope of the lines are -1 and -2

If m = $$- \dfrac{1}{2}$$, then the slope of the lines are $$- \dfrac{1}{2}$$ and -1

Case 2:

$$\dfrac{1}{3}=|\dfrac{-m}{1+2m^2}|$$

2m2 – 3m + 1 = 0

2m2 – 2m – m + 1 = 0

2m (m – 1) – 1(m – 1) = 0

m = 1 or $$\dfrac{1}{2}$$

If m = 1, then the slope of the lines are 1 and 2

If m = $$\dfrac{1}{2}$$, then the slope of the lines are $$\dfrac{1}{2}$$ and 1

The slope of the lines are [-1 and -2] or [$$- \dfrac{1}{2}$$ and -1]

or [1 and 2] or [$$\dfrac{1}{2}$$ and 1]

Answered by Abhisek | 1 year ago

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