Question

A line AB meets the x-axis at point A and y-axis at point B. The point P(−4, −2) divides the line segment AB internally such that AP : PB = 1 : 2. Find:

1. the co-ordinates of A and B.
2. equation of line through P and perpendicular to AB.

Answer 1

∵ Line AB intersects x-axis at A and y-axis at B.

i. Let co-ordinates of A be (x, 0) and of B be (0, y)

Point P(−4, –2) intersects AB in the ratio 1 : 2 internally

∴ `-4 = (1 xx 0 + 2 xx x)/(1 + 2)`  ...`(x = (m_1x_2 + m_2x_1)/(m_1 + m_2))`

`\implies -4 = (0 + 2x)/3`

`\implies` 2x = –12

∴ `x = (-12)/2 = -6` and `-2 = (1 xx y + 2 xx 0)/(1 + 2)`  ...`[∵ y = (m_1y_2 + m_2y_1)/(m_1 + m_2)]`

`\implies - 2 = (y + 0)/3`

`\implies` y = −6

∴ Co-ordinates of A will be (−6, 0) and Co-ordinates of B will be (0, −6)

ii. Now, slope of AB (m₁)

= `(y_2 - y_1)/(x_2 - x_2)`

= `(-6 - 0)/(0 - (-6))`

= `(-6)/6`

= −1

∴ Slope of the line perpendicular to AB (m₂) = 1  ...(∵ m₁m₂ = −1)

∴ Equation of line perpendicular to AB and drawn through P(−4, −2) will be

y − y₁ = m(x − x₁)

`\implies` y + 2 = 1(x + 4)

`\implies` y + 2 = x + 4

`\implies` y = x + 4 − 2

`\implies` y = x + 2

Hence, required equation of the line is y = x + 2