Question

If the equations ${{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0~$ and ${{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0~$ represent two perpendicular lines then prove that ${{a}_{1}}{{a}_{2}}+{{b}_{1}}{{b}_{2}}=0$.

Answer 1

We start solving the problems by finding the slope of the lines ${{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0~$ and ${{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0~$using the fact that slope of a line of the form $ax+by+c=0$ is given as m$=\dfrac{-a}{b}$ that is the division of the negative of the coefficient of x with coefficient of y. Now, we use the fact that if two lines are perpendicular then ${{m}_{1}}\cdot {{m}_{2}}=-1$ (Where ${{m}_{1}}$ is the slope of line 1 and ${{m}_{2}}$ is the slope of line 2) and make subsequent calculations to complete the required proof.

Complete step by step answer:
According to the problem, if the lines ${{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0~$ and ${{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0~$ are perpendicular, then we have to prove that ${{a}_{1}}{{a}_{2}}+{{b}_{1}}{{b}_{2}}=0$.
We know that the slope of the line with the equation of the form $ax+by+c=0$ is $\dfrac{-a}{b}$. Now, we use this result to find the slope of the lines ${{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0~$ and ${{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0~$.
Slope of line 1 $a{{x}_{1}}+b{{y}_{1}}+{{c}_{1}}=0$ is $\dfrac{-{{a}_{1}}}{{{b}_{1}}}$.
Similarly, the slope of line 2 $a{{x}_{2}}+b{{y}_{2}}+{{c}_{2}}$ is $\dfrac{-{{a}_{2}}}{{{b}_{2}}}$.
Now, as these two lines are perpendicular to each other, hence, using the formula given in the hint, we can write as follows
$\left( \dfrac{-{{b}_{1}}}{{{a}_{1}}} \right)\cdot \left( \dfrac{-{{b}_{2}}}{{{a}_{2}}} \right)=-1$.
$\Rightarrow \dfrac{{{b}_{1}}}{{{a}_{1}}}\cdot \dfrac{{{b}_{2}}}{{{a}_{2}}}=-1$.
$\Rightarrow \dfrac{{{b}_{1}}{{b}_{2}}}{{{a}_{1}}{{a}_{2}}}=-1$.
$\Rightarrow {{b}_{1}}{{b}_{2}}=-{{a}_{1}}{{a}_{2}}$.
$\Rightarrow {{b}_{1}}{{b}_{2}}+{{a}_{1}}{{a}_{2}}=0$.

Hence, we have proved that ${{a}_{1}}{{a}_{2}}+{{b}_{1}}{{b}_{2}}=0$.

Note: We should not confuse the formula of slope of the line of the form $ax+by+c=0$ as $\dfrac{-b}{a}$. This can be derived by converting the given line to the slope form $y=mx+c$ and comparing it with the coefficient of x in each equation. We should confuse the condition of perpendicular lines with the condition of parallel lines.