Question

prove that two lines are perpendicular if the product of their slope is -1​

Answer 1

$\huge\bf{\red{\overbrace{\underbrace{\purple{Given:}}}}}$

★Two Non-vertical lines.

$\huge\bf{\red{\overbrace{\underbrace{\purple{To\:\:Prove:}}}}}$

★The two lines are perpendicular.

$\huge\bf{\red {\overbrace{\underbrace{\purple{Concept\:\:Used:}}}}}$

★We will trigonometry to formula of slope.

$\huge\bf{\red {\overbrace{\underbrace{\purple{Proof:}}}}}$

Let

→ $L_{1} \:and\:L_{2} \:be\:two\:non-parallel\:lines.$

→$\theta_{1}$ & $\theta_{2}\:be\:their\:inclinations\: respectively$

Now, we know,

$\large\green{\boxed{\bf{\red{Slope=\dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}=tan\theta}}}}$

So,

Slope of line $L_{1}$ = tan$\theta_{1}$

${\underline{\boxed{\red{.\degree.m_{1}=tan\theta_{1}}}}}$

Slope of line $L_{2}$ = tan$\theta_{2}$

${\underline{\boxed{\red{.\degree.m_{2}=tan\theta_{2}}}}}$

______________________________________

If $L_{1}||L_{2}$ then,

$\implies \theta_{2}=90\degree +\theta_{1}$

(by exterior angle property)

$\implies tan\theta_{2}=tan(90+\theta_{1})$

(Using tan both sides)

$\implies tan\theta_{2}=-cot\theta_{1}$

$\large\purple {\boxed{\bf{\pink{ tan(90+\alpha) =-tan\alpha}}}}$

$\implies tan\theta_{2}=\dfrac{-1}{tan\theta_{1}}$

$\implies tan\theta_{1}\tan\theta_{2}=-1$

${\underline{\boxed{\orange{.\degree.m_{1}m_{2}=-1}}}}$

Hence proved.

Answer 2

Step-by-step explanation:

$hope \: it \: will \: help \: u$