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Let AB be a line segment of length 4 wit...

Let AB be a line segment of length 4 with A on the line `y = 2x` and B on the line `y = x`. The locus of the middle point of the line segment is

A

`25x^(2)+13y^(2)+36xy-4=0`

B

`25x^(2)+13y^(2)-36xy-4=0`

C

`13x^(2)+25y^(2)-36xy-4=0`

D

`13x^(2)+25y^(2)+36xy-4=0`

Text Solution

Verified by Experts

The correct Answer is:
B

Let `B=(alpha, alpha)` and middle point AB is (h, k)
Then, `A-= (2h-alpha, 2k-alpha)`
Lies on `y=2x`
Then, `(2k-alpha)=2(2h-alpha)" " therefore alpha=4h-2k`
`|AB|=4 rArr sqrt((2h-2alpha)^(2)+(2k-2alpha))=4`
or `(h-alpha)^(2)+(k-alpha)^(2)=4 " or " [h-(4h-2k)]^(2)+[k-(4h-2k)]^(2)=4`
`rArr (-3h+2k)^(2)+(-4h+3k)^(2)=4" or "25h^(2)+13k^(2)-36hk=4`
Required locus is `25x^(2)+13y^(2)-36xy-4=0`
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