The area of the triangle formed by the coordinate axes and a line is 6 square units and the length of the hypotenuse is 5 units . Find the equation of the line

Question

The area of the triangle formed by the coordinate axes and a
line is 6 square units and the length of the hypotenuse is 5 units
Find the equation of the line

ramandeep.singh · Accepted Answer

Let the line cut the x-axis and the y-axis at A(a, 0) and B(0, b) respectively.
∴OA = a

OB = b
Area of the ΔOAB = $\frac{1}{2} a b$ =6

=> ab = 12

AB is the hypotenuse of the right ΔOAB.
$A B^{2} = a^{2} + b^{2} = 25 (a + b)^{2} = a^{2} + b^{2} + 2 a b = 25 + 2 (12) = 49 ∴ a + b = \pm 7.... (1)$

$(a - b)^{2} = a^{2} + b^{2} - 2 a b = 25 - 2 (12) = 1 ∴ a - b = \pm 1.... (2)$
From equations (1) and (2) , we get

a = 4

b = 3
Hence, the equation of the line AB is

$\frac{x}{4} + \frac{y}{3} = 1 \Rightarrow 3 x + 4 y = 12$
If we take the negative values of the square roots in (1) and (2), we get

a = - 4

b = - 3
then the equation of the line AB is

$\frac{x}{- 4} + \frac{y}{- 3} = 1 \Rightarrow 3 x + 4 y = - 12$