IIT JEE , AIEEE Forum - Mathematics , Algebra

Radon222

The altitude through A of triangle ABC meets BC at D(2,3) and the circumscribed circle at E(5,5).The ordinate of orthocentre being a natural number,what is the probability that orthocentre lie on the lines
                                                y=1
                                                y=2
                                                y=3
                                              .........
                                              .........
                                                y=10 ?

Decoder

the altitude lines passes through thse pts..i.e

(2,3) & (5,5)
now obtain the equation of this line..

line is y= {2x+ 5 } / 3 ..

this can be written as.. y = {2(x+1) + 3} / 3

so we want x+1 to be divisible by 3..

now it can be 2,5,8,11,14 ...(A.P = 3n-1) where n

N.

now putting values of each in this equation..u get (y=a) type of line...

they also follow a pattern i.e ( 2n+1) where n

N..

so in given domain only..3,5,7,9 ..r included so probalility..is 2/5..

i think its fine..but better if a finer method is presented..

abhishekray07

does the orthocenter lie on y=1?
that means probability is 1/10?

Radon222

answer is 3/5

Decoder

it will be in the case of x=a ..lines..
i think i have commited some mistake in last steps..otherwise everything is fine..

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