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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.
Solution
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
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