Question

Find the area of the triangle formed by the line y = m₁ x + c₁, y = m₂ x + c₂ and x = 0.

Answer 1

y = m₁x + c₁      ... (1)
y = m₂x + c₂          ... (2)
x = 0                      ... (3)
In triangle ABC, let equations (1), (2) and (3) represent the sides AB, BC and CA, respectively.
Solving (1) and (2):

\[x = \frac{c_2 - c_1}{m_1 - m_2}, y = \frac{m_1 c_2 - m_2 c_1}{m_1 - m_2}\]

Thus, AB and BC intersect at B \[\left( \frac{c_2 - c_1}{m_1 - m_2}, \frac{m_1 c_2 - m_2 c_1}{m_1 - m_2} \right)\].

Solving (1) and (3):

\[x = 0, y = c_1\]

Thus, AB and CA intersect at A \[\left( 0, c_1 \right)\].

Similarly, solving (2) and (3):

\[x = 0, y = c_2\]

Thus, BC and CA intersect at C \[\left( 0, c_2 \right)\].

∴ Area of triangle ABC = \[\frac{1}{2}\begin{vmatrix}0 & c_1 & 1 \\ 0 & c_2 & 1 \\ \frac{c_2 - c_1}{m_1 - m_2} & \frac{m_1 c_2 - m_2 c_1}{m_1 - m_2} & 1\end{vmatrix}\]    = \[\frac{1}{2}\left( \frac{c_2 - c_1}{m_1 - m_2} \right)\left( c_1 - c_2 \right) = \frac{1}{2}\frac{\left( c_1 - c_2 \right)^2}{m_2 - m_1}\]