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Q.

The intercept on the x-axis made by the tangent to the curve, y= 0 x t dx ,xR  , which are parallel to the line y = 2x, are equal to:


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a

±2  

b

±3  

c

±4  

d

±1   

answer is D.

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Detailed Solution

We know that the given curve is  y= 0 x t dx   and we also know that the differentiation of any curve gives the general tangent equation.
So, after differentiating the curve,  y= 0 x t dx  , we will get:
dy dx = 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 dy dx = x =2  .
So x=±2  
Now, we know that y= 0 x t dx  , so when x = 2.
y= 0 2 t dt y= 0 2 tdt y= t 2 2 2 =2  
Also, when x = -2, then:
y= 0 2 t dt y= 0 2 tdt 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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The intercept on the x-axis made by the tangent to the curve, y= ∫ 0 x t dx ,x∈R  , which are parallel to the line y = 2x, are equal to: