Question

If the straight line $ax + by + c = 0$ always passes through $\left( {1, - 2} \right)$, then $a,b,c$ are:

1. In A.P.
2. In H.P.
3. In G.P.
4. None of these

Answer 1

In order to solve this question, we will substitute $1$ in place of $x$ and $- 2$ in place of $y$ in the equation of line $ax + by + c = 0$. After that, we will add $2b$ each side of the obtained expression and cancel out the equal-like term to simplify the obtained expression.

Complete step-by-step solution:
Since, the given equation of line is $ax + by + c = 0$ and the given point is $\left( {1, - 2} \right)$ from which the line passes.
If the line passes through any point, the point will satisfy the equation of the line.
Here, we will substitute $1$ in place of $x$ and $- 2$ in place of $y$ in the equation of line $ax + by + c = 0$ as:
$\Rightarrow 1 \times a + \left( { - 2} \right)b + c = 0$
Now, we will simplify the equation using multiplication with variables.
$\Rightarrow a - 2b + c = 0$
Then, we will add $2b$ each side of the obtained expression and will simplify it.
$\Rightarrow a - 2b + c + 2b = 0 + 2b$
Here, we will cancel out the term $2b$ from the left side of obtained expression and simplify it using addition and subtraction.
$\Rightarrow a + c = 2b$
Now, we will divide by $2$ in the obtained expression and simplify it.
$\Rightarrow \dfrac{{a + c}}{2} = \dfrac{{2b}}{2}$
After that, we will cancel out the term $2$ from the right side of the obtained expression and simplify it.
$\Rightarrow \dfrac{{a + c}}{2} = b$
Since, the obtained expression denotes that $a,b,c$ are in A.P, the correct option is 1.

Note: A.P. is an order of a series of a number in which the mid term is the average of the first and last term. Let, $a,b,c$ are a series in A.P of three numbers in which $b$ is mid term. Then, $\Rightarrow b = \dfrac{{a + c}}{2}$.