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Question

The line y=mx intersects the circle x2+y24x4y=0 and x2+y2+6x8y=0 at points AB (points being other than the origin). The range of m such that the origin divides AB internally is


A
m>43 or m<2
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B
m>1
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C
2<m<43
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D
1<m<34
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Solution

The correct option is D 1<m<34

O divides AB internally in the ratio λ:1
λ>0
A(x1,y1),B(x2,y2),y=mx
A(x1,mx1),B(x2,mx2)
Section formula : (0,0)=(λx2+x1λ+1,λmx2+mx1λ+1)
x1=λx2
Or λ=x1x2>0x1x2<0

x21+m2x214x14mx1=0;x22+m2x22+6x28mx2=0

x1=4(1+m)1+m2;x2=2(4m3)1+m2
x1.x2<0
(1+m)(4m3)<0
Critical points m=1,34
1<m<34
 

Mathematics

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