Question

The line segment joining the points A(3, −4) and B(−2, 1) is divided in the ratio 1 : 3 at point P in it. Find the co-ordinates of P. Also, find the equation of the line through P and perpendicular to the line 5x – 3y = 4.

Answer 1

Point P, divides the line segment A(3, −4) and B(−2, 1) in the ratio of 1 : 3.

Let co-ordinates of P be (x, y), then

`x = (m_1x_2 + m_2x_1)/(m_1 + m_2)`

= `(1 xx (-2) + 3 xx 3)/(1 + 3)`

= `(-2 + 9)/7`

= `7/4`

`y = (m_1y_2 + m_2y_1)/(m_1 + m_2)`

= `(1 xx (1) + 3(-4))/(1 + 3)`

= `(1 - 12)/4`

= `(-11)/4`

∴ Co-ordinates of P are `(7/4, (-11)/4)`

Writing the line 5x – 3y = 4 in the form of y = mx + c

`\implies` −3y = −5x + 4

`\implies y = 5/3 x - 4/3`

∴ `m = 5/3`

And slope of the line perpendicular to it

= `-(3/5)`

= `-3/5`  ...(∵ Product of slopes = –1)

∴ Equation of the requird line is given by y − y₁ = m(x − x₁)

`\implies y - ((-11)/4) = (-3)/5(x - 7/4)`

`\implies y + 11/4 = (-3)/5 = (x - 7/4)`

`\implies 5y + 55/4 = -3x + 21/4`

`\implies 3x + 5y = 21/4 - 55/4`

= `(-34)/4`

= `(-17)/2`

`\implies` 6x + 10y = –17

`\implies` 6x + 10y + 17 = 0