Question

The foot of the perpendicular drawn from the origin, on the line, 3x + y = λ(λ ≠ 0) is P. lf the line meets x-axis at A and y-axis at B, then the ratio BP : PA is ______.

पर्याय

• 9 : 1

• 1 : 3

• 1 : 9

• 3 : 1

Answer 1

The foot of the perpendicular drawn from the origin, on the line, 3x + y = λ(λ ≠ 0) is P. lf the line meets x-axis at A and y-axis at B, then the ratio BP : PA is 9 : 1.

Explanation:

Given: 3x + y = λ

From the given data, we plot the graph as shown in the following figure:

Slope of line = −3 = tan θ

Now, from ΔOAB: cos (π − θ) = `(AP)/(OA)`

cos θ = `(-3AP)/(lambda) ⇒ AP = (-lambda)/(3) cos θ`

Now, from ΔOBP:

cos(θ − π/2) = `(BP)/(OB)`

sin θ = `(BP)/(lambda)` ⇒ BP = λ sin θ

`(BP)/(AP) = (- lambda sin theta)/((lambda/3 cos theta)) = -3 (sin theta)/(cos theta)`

= −3tan θ = `9/1`