Question

If the line 3x + 4y = p makes a triangle of area 24 square units with the co-ordinate axes, then find the value of p.

Answer 1

Let the line 3x + 4y = p cuts the X and Y-axes at points A and B respectively.
3x + 4y = p

∴ `(3x)/"p" + (4y)/"p"` = 1

∴ `x/("p"/3) + y/("p"/4)` = 1

This equation is of the form

`"x"/"a" + y/"b"` = 1.

with a = `"p"/3 and "b" = "p"/4`

∴ A ≡ `("a", 0) = ("p"/3 , 0) and "B" ≡ (0, "b") = (0, "p"/4)`

∴ OA = `"p"/3 and "OB" = "p"/4`

Given, A (Δ.OAB) = 24 sq. units

∴ `1/2 xx "OA" xx "OB"` = 24

∴ `1/2 xx "p"/3 xx "p"/4`= 24

∴ p² = 576
∴ p = ± 24