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Two lines are drawn at right angles, one...

Two lines are drawn at right angles, one being a tangent to `y^2=4a x` and the other `x^2=4b ydot` Then find the locus of their point of intersection.

Text Solution

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`y=mx+(a)/(m)` (1)
is a tangent to `y^(2)=4ax` and
`x=m_(1)y+(b)/(m_(1))` (2)
is a tangent to `x^(2)=4by`.
Lines (1) and (2) are perpendicular. Therefore,
`mxx(1)/(m^(1))=-1`
`or" "m_(1)=-m`
Let (h,k) be the point of intersection of (a) and (2). then,
`k=mh+(a)/(m)andh=m_(1)k+(b)/(m^(1))`
`or" "k=mh+(a)/(m)andh=-mk-(b)/(m)`
`or" "k^(2)h-mk+a=0andm^(2)k+mh+b=0`
`or(m^(2))/(-kb-ah)=(m)/(ak-bh)=(1)/(h^(2)+k^(2))`
[By cross-multiplication]
Eliminating m, we have
`-(h^(2)+k^(2))(kb+ah)=(bh-ak)^(2)`
`or" "(x^(2)+y^(2))(ax+by)+(bx-ay)^(2)=0`
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