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

Line AB passes through the points A(1, 0) and B(2, 3).

∴ Slope of AB = `(3 - 0)/(2 - 1) = 3/1`

PQ ⊥ AB

Slope of AB = `3/1`

∴ Slope of PQ, m = `- 1/(3/1) = -1/3`

PQ intersects line AB at C.

Also, point C divides the line segment AB in the ratio 1 : n.

That is `"C"((1 xx 2 + "n" xx 1)/("n" + 1), (1 xx 3 + "n" xx 0)/("n" + 1))`

or `"C" (("n" + 2)/("n" + 1), 3/("n" + 1))`

Now the equation of line PQ,

`"y" - "y"_1 = "m"("x" - "x"_1)`

Where, x₁ = `("n" + 2)/("n" + 1)` and y₁ = `3/("n" + 1)`

`"y" - 3/("n" + 1) = -1/3("x" - ("n" + 2)/("n" + 1))`

3(n + 1)y − 9 = −[(n + 1)x − (n + 2)]

or (n + 1)x + 3(n + 1)y = n + 2 + 9 = n + 11

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