Verify that the equation of a line passing through the point of intersection of two lines aix+ by +c=0 and a2x + b₂y+c2=0 is of the form

To verify that the equation of a line passing through the point of intersection of two lines is of a specific form, let's denote the two lines as follows:

1. Line 1: $a_1x + b_1y + c_1 = 0$
2. Line 2: $a_2x + b_2y + c_2 = 0$

1. ### Break Down the Problem

We want to derive the equation of a line using the point of intersection of the given lines.

2. ### Relevant Concepts

The point of intersection $(x_0, y_0)$ of the two lines can be found using determinants. The general form of the equation for a line can be expressed in terms of the coefficients of the two given lines.

3. ### Analysis and Detail

To find the point of intersection:

$\begin{vmatrix} x & y & 1 \\ a_1 & b_1 & -c_1 \\ a_2 & b_2 & -c_2 \end{vmatrix} = 0$

Expanding this determinant gives us the coordinates of the intersection.

Let the equation of the new line through the intersection point be represented as:

$\lambda (a_1x + b_1y + c_1) + \mu (a_2x + b_2y + c_2) = 0$

Where $\lambda$ and $\mu$ are parameters.

By taking specific values for $\lambda$ and $\mu$ (e.g., $\lambda = 1$ and $\mu = 1$), we get:

$(a_1 + a_2)x + (b_1 + b_2)y + (c_1 + c_2) = 0$

Thus, this confirms that the equation of a line passing through the intersection is indeed a linear combination of the two original lines.

4. ### Verify and Summarize

The derived equation maintains the general form expected for any line:

$A x + B y + C = 0$

where $A = a_1 + a_2$, $B = b_1 + b_2$, and $C = c_1 + c_2$. Thus, we have verified that the equation of the line that passes through the intersection of the two lines is of the form:

$a x + b y + c = 0$

Final Answer

Hence, the equation of a line passing through the point of intersection of the two lines is confirmed to be of the form $a x + b y + c = 0$.