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Question

The intercepts on x-axis made by tangents to curve, y=x0|t|dt, xR, which are parallel to the line y=2x, are equal to:
  1. ±1
  2. ±2
  3. ±3
  4. ±4


Solution

The correct option is A ±1
y=2x...(1)
m1=dydx=2
Now, y=x0|t|dt, xR
Case1: if t>0
y=x0tdt
y=x22...(2)
m2=dydx=2x2=x
As, m1=m2x=2
From equation (2),y=2
The equation of line pasing through (2,2) and have slope 2 is
y2=2(x2)
It has intercept on x axis, so, y=0
02=2(x2)x=1
intercept on x-axis =1
Case2: If t<0
y=x0tdt
y=x22...(3)
m3=dydx=2x2=x
As, m3=m1x=2
From equation (3),y=2
The equation of line pasing through (2,2) and have slope 2 is
y+2=2(x+2)
It has intercept on x axis, so put, y=0
0+2=2(x+2)x=1
intercept on x-axis =1
Hence, from case1 and case2, intercept on x-axis =±1

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