The perpendicular from the origin to a line meets it at the point (–2, 9), find the equation of the line.

Points are origin (0, 0) and (-2, 9).

We know that slope, m = 

\( \dfrac{y_2+y_1}{x_2+x_1}\)

=\( \dfrac{9-0}{2-0}\)

= \(- \dfrac{9}{2}\)

We know that two non-vertical lines are perpendicular to each other if and only if their slopes are negative reciprocals of each other.

m = (\( - \dfrac{1}{m}\)) = \( \dfrac{-1}{(-\dfrac{9}{2})}\) = \( \dfrac{2}{9}\)

We know that the point (x, y) lies on the line with slope m through the fixed point (x₀, y₀), if and only if, its coordinates satisfy the equation y – y₀ = m (x – x₀)

y – 9 = (\( \dfrac{2}{9}\)) (x – (-2))

9(y – 9) = 2(x + 2)

9y – 81 = 2x + 4

2x + 4 – 9y + 81 = 0

2x – 9y + 85 = 0

The equation of line is 2x – 9y + 85 = 0.