If the locus of the circumcentre of a variable triangle having sides y−axis, 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 D18a In the above diagram, C≡(0,1m) B=(1−2ml,2) since from the figure it is clear required triangle will be a rightangle triangle. Let (h,k) be the circumcentre of △ABC. ∴h=1−2m2l, k=1+2m2m⇒m=12(k−1) ∴l=k−22h(k−1),
∵(l,m) lies on y2=4ax ∴m2=4al ⇒(12(k−1))2=4a{k−22h(k−1)} h=8a(k2−3k+2) ∴ Locus of (h,k) is x=8a(y2−3y+2) ⇒(y−32)2=18a(x+2a) ∴ Vertex is (−2a,32). ∵ Length of smallest focal chord = Length of latusrectum =18a units