Question

A line perpendicular to the line segment joining the points (1, 0) and (2, 3) divides it in the ratio 1:n. Find the equation of the line.

Answer 1

Given:

Points A(1, 0) and B(2, 3)
The line divides AB internally in the ratio 1 : n
The required line is perpendicular to AB

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Step 1: Find the slope of AB

Slope of AB =

m_AB = (3 − 0) / (2 − 1) = 3

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Step 2: Find the slope of the required line

If two lines are perpendicular:

m₁ · m₂ = −1

∴ Required slope =

m = −1 / 3

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Step 3: Find the point dividing AB in the ratio 1 : n

Using section formula:

Coordinates of point P =

( (1·2 + n·1)/(1 + n) , (1·3 + n·0)/(1 + n) )

P = ( (2 + n)/(n + 1) , 3/(n + 1) )

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Step 4: Equation of the required line

Using point–slope form:

y − 3/(n + 1) = (−1/3)[ x − (2 + n)/(n + 1) ]

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Step 5: Simplify

3y − 9/(n + 1) = −x + (2 + n)/(n + 1)

3y = −x + (n + 11)/(n + 1)

x + 3y = (n + 11)/(n + 1)

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Final Answer:

The equation of the required line is:
x + 3y = (n + 11)/(n + 1)