Question

the Slope of a line is double of the slope of another line if tangent of the angle between them is 1 by 3 find the slope of the lines​

Answer 1

Answer:

This is the answer

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Answer 2

Given :-

Slope of a line = Double of the slope of another line

Tangent of the angle between them = 1/3

To Find :-

Slope of the lines.

Solution :-

Let us consider ‘$\sf m_1$’ and ‘m’ be the slope of the two given lines such that $\sf m_1$ = 2m

We know that if θ is the angle between the lines l1 and l2 with slope $\sf m_1$ and $\sf m_2$, then

$\sf tan \ \theta =\left| \dfrac{(m_2-m_1)}{(1+m_1 m_2)} \right|$

Given that,

Tangent of the angle between them = 1/3

$\sf \dfrac{1}{3} = \left| \dfrac{m-2m}{1+2m \times m} \right| = \left| \dfrac{-m}{1+2m^{2}} \right|$

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

CASE I :

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

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

$\sf 2m^{2} +1 +3m = 0$

$\sf 2m (m+1) + 1(m+1) = 0$

$\sf (2m+1) (m+1)= 0$

$\sf m = -1$

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

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

CASE II :

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

$\sf 2m^{2} - 3m + 1 = 0$

$\sf 2m^{2} - 2m - m + 1 = 0$

$\sf 2m (m - 1) - 1(m - 1) = 0$

$\sf m = 1$

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

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

∴ The slope of the lines are [-1 and -2] or [-1/2 and -1] or [1 and 2] or [1/2 and 1]