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Question

If the locus of the circumcentre of a variable triangle having sides yaxis, y=2 and lx+my=1, where (l,m) lies on the parabola y2=4ax is a curve C, then the length of smallest focal chord of this curve C (in units) is

A
112a
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B
14a
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C
116a
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D
18a
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Solution

The correct option is D 18a
In the above diagram,
C(0,1m)
B=(12ml,2)
since from the figure it is clear required triangle will be a rightangle triangle.
Let (h,k) be the circumcentre of ABC.
h=12m2l,
k=1+2m2mm=12(k1)
l=k22h(k1),

(l,m) lies on y2=4ax
m2=4al
(12(k1))2=4a{k22h(k1)}
h=8a(k23k+2)
Locus of (h,k) is x=8a(y23y+2)
(y32)2=18a(x+2a)
Vertex is (2a,32).
Length of smallest focal chord
= Length of latusrectum =18a units

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