# What are the values of a and b if the area of the triangle formed by the lines ax + by + 4 = 0, and the x and y axis is equal to 24.

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The triangle is made up of the line ax + by + 4 = 0, the x – axis and the y-axis.

Now the x and y axes are perpendicular.

The equation of the third line ax + by + 4 = 0 can be written as -ax/4 + -by/4 = 1. Here the x intercept is –4/ a and the y intercept is –4/ b.

As the area of the triangle is 24, we have 24 = (1/2)* base * height . Substituting the values of base and height as teh intercepts on the axes we get (1/2)*(-4/a)*(-4/b) = 24

=> 16/ ab = 48

=> ab = 16 / 48

=> ab = 1/3

Therefore the values of a and b are not unique but a has to be equal to 1/3b.

ax+by+4 = o intercepts x-axis at -4/a and y-axis at -4/b.

Therefore the area A of the right angle formed by the x intercept -4/a and the y intercept -4/b with a right angle at the origin is given by:

A= -4/b is (1/2)(x intercept)*(y intercept).

A = (1/2)(-4/a)((-4/b).

The area A is actually given to be 24.

Therefore (1/2){16/ab} = 24.

=> 8/ab = 24.

=> 8/24 - ab.

Therefore the values of a and b are any value that satisfy ab = 1/3. Or 3ab = 1.

=> ab = 1/3.