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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 + 13 y^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, 2 k -alpha)`
Lies on `y = 2x`
Then, `(2k -alpha) = 2 (2h - alpha) :. alpha = 4h - 2k`
`abs(AB) = 4 rArr sqrt((2h - 2alpha)^2 + (2k - 2alpha)^2) =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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