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Question

If the angle between the lines joining origin and the point of intersection of the line xy=2 and the curve x24xy+2y22x+y=0 is 45 then find k.

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Solution

substitute y=mx in xy=2

we get,

x=21m,y=2m1m

substitute these values in equation, x24xy+2y22x+y+k=0

(21m)2421m2m1m+2(2m1m)2221m+2m1m+k=0

solving the above equation, we get,

(k+6)m2(2k+10)m+k=0..........(1)

m=k+5±4k+25k+6

m1m2=24k+25k+6..........(2)

m1m2=kk+6........(3)

since given,

lines making an angle of 45

m1m21+m1m2=1

(m1m2)2=(1+m1m2)2

From (2) and (3)

(24k+25k+6)2=(1+kk+6)2

solving the above equation we get

k=1±17

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