For which of the following values of m is the area of the region bounded by the curve $y = x − x 2$ and the line $y = m x$ equal to $92$ .

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-2

answer is B.

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Detailed Solution

Let the equation of the straight line be

y = m x

… (1), and the given equation of the curve be

y = x − x 2

… (2).
Given that the area of the region between the curve and the line is

92

… (3)
To find the point of intersection, substitute the value of y from (1) into (2) to get:

m x = x − x 2 x 2 + m x − x = 0 x {x + (m − 1)} = 0 x = 0, (1 − m)

In the above solution, m can be either positive or negative.
Case 1: m is negative (i.e.)

m < 0

So, the straight line has a negative slope. The graph of the curve and line is given by,
seo

As the straight line is below the x-axis, so the area of the straight line is negative. So, the area of the region between the curve and the line is given by,
Question Image

Comparing the above value with (3), we get

(1 − m) 3 6 = 92 (1 − m) 3 = 27 1 − m = 3 1 − 3 = m m = − 2

Case 2: m is positive (i.e.)

m > 0

So, the straight line has a positive slope. The graph of the curve and line is given by,
seo

As the straight line is below the x-axis, so the area of the straight line is negative. So, the area of the region between the curve and the line is given by,
Question Image

Comparing the above value with (3), we get