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Question

The slope of the line touching both the parabolas y2=4xandx2=-32y is


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Solution

Compute the slope of the line touches parabola:

x2=โˆ’32y,y2=4x
m be the slope of the common tangent
Equation of tangent of parabola y2=4ax

y=mx+am

Here, a=1

y=mx+1m...(i)
Equation of tangent of parabola x2=-4ay
y=mx+am2

y=mx+8m2...(ii)
(i)and(ii) are identical

1m=8m2โ‡’m3=18m=12

Alternate method for computing the slope of the line

Let tangent toy2=4x be y=mx+1m
as this is also tangent to x2=-32y
Solving x2+32mx+32m=0
Since roots are equal

โˆดD=0โ‡’(32)2โˆ’4ร—32m=0โ‡’m3=432โ‡’m=12

The slope of the line touching both the parabolas is m=12

Hence, the slope of line touching both the parabolas is m=12


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