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Question
If the line 3x + 4y = p makes a triangle of area 24 square unit 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
∴ `(3"x")/"p" + (4"y")/"p"` = 1
∴ `"x"/("p"/3) + "y"/("p"/4)` = 1
This equation is of the form `"x"/"a" + "y"/"b"` = 1,
where 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(ΔOAB) = 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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