Question

a student said that the line join by p(-2,1)and Q(2,5) subtend an angle with x axis is 45°.do you agree. justify your answer
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Answer 1

Formula.

If $m_{1}$ and $m_{2}$ be the slopes of two lines and $\theta$ be the angle between them, then

$\quad tan\theta=\frac{m_{1}\sim m_{2}}{1+m_{1}m_{2}}$

Solution.

The slope of the straight line $PQ$ which passes through the points $P\:(-2,\:1)$ and $Q\:(2,\:5)$ is

$\quad m_{1}=\frac{5-1}{2-(-2)}$

$\Rightarrow m_{1}=\frac{4}{4}$

$\Rightarrow \color{blue}{m_{1}=1}$

The slope of the $x-$ axis is

$\quad \color{blue}{m_{2}=0}$

Then the angle between them is

$\quad tan^{-1}\left(\frac{1-0}{1+0}\right)$

$=tan^{-1}(1)$

$=\color{blue}{45^{\circ}}$

Answer.

$\quad\quad\color{blue}{\text{Yes, the student was right.}}$

Answer 2

Given:

A student said that the line joined by p(-2,1)and Q(2,5) subtend an angle with x axis is 45°.

To find:

Do you agree. justify your answer.

Solution:

From given, we have,

p(-2, 1) and Q(2, 5)

in order to find the slope of line formed by these lines,  we use 2 point form given by,

x₁ = -2, y₁ = 1 and

x₂ = 2, y₂ = 5

$m = \dfrac{y_2-y_1}{x_2-x_1}$

$m_1 = \dfrac{5-1}{2-(-2)}=\dfrac{4}{4}$

∴ m₁ = 1

x axis

the slope of line formed by x axis is zero.

∴ m₂ = 0

The angle formed by 2 lines given by a formula,

$tan \theta = \dfrac{m_1-m_2}{1+m_1m_2}$

$tan \theta = \dfrac{1-0}{1+1 \times 0}=\dfrac{1}{1}=1$

θ = tan^-1 1

∴ θ = 45°

Yes, the line joined by p(-2, 1) and Q(2, 5) subtend an angle with x axis is 45°.