Arun
Last Activity: 5 Years ago
Let the equation of the parabola
=>y^2=4ax
So that it is directrix is is x+a=0
We need to show that the x-coordinate of the orthocentre is -a
Assume any three point s x1,x2 and x3
Intersections of the three tangents to the parabola on these three points will be
=>P(ax1x2,a(x1+x2))
=>Q(ax2x3,a(x2+x3))
=>R(ax3x1,a(x3+x1))
The slope of QR=[a(x2+x3)-a(x3+x1)]/[ax2x3-ax3x1]=1/x...
There fore the equation of the altitude through P is
=>y-a(x1+x2)=-x3(x-ax1x2)
=>y+x3x=a(x1+x2+x1x2x3)......(1)
The equation of the altitude through Q is onto PR
=>y+x1x=a(x2+x3+x1x2x3)..........(2)
By equation (1)-(2)
=>(x3-x1)x=a(x1-x3)
=>x=-a
Hence the orthocentre of the triangle formed by any three tangents to the parabola is on its
directrix
