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

Hint: 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.