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Question

The slope of the line touching both the parabolas y2=4x and x2=32y is :

A
12
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B
32
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C
18
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D
23
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Solution

The correct option is A 12
Let slope of tangent be m


Equation of tangent in terms of slope m for parabola y2=4x is
y=mx+1m(1)
This is also a tangent to x2=32y, so
x2=32(mx+1m)mx2+32m2x+32=0
From the condition of tangency, we get
D=0(32m2)24×m×32=032m44m=04m(8m31)=0m=0 or m=12
m=0 (not possible because it is not a tangent to parabola y2=4ax)
m=12


Alternate solution :
Equation of tangent in terms of slope m for parabola y2=4x is
y=mx+1m(1)
Now for this line to be tangent to parabola x2=32y
1m=8m2
m3=18
m=12

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