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Two vertices of a triangle are (1, 3) an...

Two vertices of a triangle are (1, 3) and (4, 7). The orthocentre lies on the line `x + y = 3`. The locus of the third vertex is

A

`x^(2) - 2xy +2y^(2) - 3x - 4y +36 = 0`

B

`2x^(2)-4xy +3y^(2) - 4x - y +42 = 0`

C

`3x^(2)+xy - 4y^(2) - 2x +24y - 40 = 0`

D

`x^(2) - 4xy +3y^(2) - 2x - y +40 = 0`

Text Solution

Verified by Experts

The correct Answer is:
C

Orthocenter lies on the line `x +y = 2`
Let orthocenter be `O(alpha,3 -alpha)`
`BO _|_ AC`
`rArr (y-7)/(x-4) (alpha)/(1-alpha) =-1`
`rArr (alpha-1)/(alpha) =(y-7)/(x-4)`
`rArr 1-(1)/(alpha)=(y-7)/(x-4)`
`rArr (1)/(alpha)=1-(y-7)/(x-4)`
`rArr alpha =(x-4)/(x-y+3)`
Also, `AO _|_ BC`
`rArr (y-3+alpha)/(x-alpha) xx (4)/(3) =-1`
`rArr 4y =12 +4alpha =-3x +3 alpha`
`rArr alpha =-3x - 4y +12`
`:.` The locus is `(x-4)/(x-y+3) =- 3x -4y +12`
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