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Question

Let L be a line passing through the point of intersection of the lines x+2y+1=0 and 2x+3y1=0. The locus of the circumcentre of the triangle formed by L and coordinate axes is

A
6x10y+4xy=0
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B
6x10y4xy=0
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C
6x+10y4xy=0
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D
6x+10y+4xy=0
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Solution

The correct option is A 6x10y+4xy=0
Equation of line L is
(x+2y+1)+λ(2x+3y1)=0(1+2λ)x+(2+3λ)y+(1λ)=0
The triangle formed is rightangle so the cirmcumcentre is the midpoint of hypotenuse
Let the midpoint be (h,k)
Now, the intercepts of the line is
x intercept is λ11+2λy intercept is λ12+3λ
Now, the midpoint is
(h,k)=(λ12(1+2λ),λ12(2+3λ))2h=λ11+2λλ=2h+14h1
Similarly,
λ=4k+16k1
Now, the locus is
2h+14h1=4k+16k112hk2h+6k1=16hk+4h4k16x10y+4xy=0

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