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Question

If the parabolas y2=4b(xc) and y2=8ax have a common normal, then which one of the follwing is a valid choice for the ordered triplets (a,b,c)?

A
(12,2,0)
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B
(1,1,3)
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C
(1,1,0)
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D
(12,2,3)
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Solution

The correct option is B (1,1,3)
If m is the slope of the common normal, then the equation of normal to parabola y2=4b(xc) is given by
y=m(xc)2bmbm3 (1)

Equation of normal to parabola y2=8ax is
y=mx4am2am3 (2)
From equations (1) and (2), we get
m(xc)2bmbm3=mx4am2am3
m2(2ab)=c2(2ab)
m2=c2ab2m20c2ab20
c2ab2 (3)
Hence (1,1,3) satisfies equation (3)

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