The intercept on the x-axis made by the tangent to the curve, $y = ∫ 0 x t d x, x ∈ R$ , which are parallel to the line y = 2x, are equal to:

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± 2

± 3

± 4

± 1

answer is D.

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Detailed Solution

We know that the given curve is

y = ∫ 0 x t d x

and we also know that the differentiation of any curve gives the general tangent equation.
So, after differentiating the curve,

y = ∫ 0 x t d x

, we will get:

⇒ d y d x = x

So, the slope of the tangent at any point (x, y) to the given curve is equal to

x

.
Also, we know that the required tangent is parallel to line y = 2x and since the line is in the form of
y = mx where ‘m’ is the slope of the line.
So, we can say that the slope of the line y = 2x is equal to 2.
Then, we say that

d y d x = x = 2

.
So

x = ± 2

Now, we know that

y = ∫ 0 x t d x

, so when x = 2.

⇒ y = ∫ 02 t d t ⇒ y = ∫ 02 t d t ⇒ y = t 2 2 2 = 2

Also, when x = -2, then:

⇒ y = ∫ 0 − 2 t d t ⇒ y = ∫ 0 − 2 − t d t ⇒ y = − t 2 2 2 = − 2

So, equation of line with slope m = 2 and passing through point (2, 2) is given by:
⇒ (y − 2) = 2(x − 2)
⇒ y = 2x − 2
And, the equation of the line with slope m = 2 and passing through the point (-2, -2) is given by:
⇒(y + 2) = 2(x + 2)
⇒ y = 2x + 2
So, the intercept made by the tangent y = 2x + 2 on the x- axis is equal to:
x = - 1
And, the intercept made by the tangent y = 2x – 2 on the x-axis is equal to:
x = 1
So, x = ±1.

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