Let equation of the parabola is, y2=4ax
Tangent at any point (x1,y1) is given by yy1=2a(x+x1)
Let any two points on the parabola be (at12,2at1),(at22,2at2)
Then equation of the tangent at (at12,2at1) is given by,
2at1y=2a(x+at12)
t1y=(x+at12)⟹y=t11(x+at12)
Equation of the tangent at point (at22,2at2) is given by
2at2y=2a(x+at22)
t2y=(x+at22)⟹y=t21(x+at22)
Since, these these tangents are perpendicular then,
t11t21=−1⟹t21=−t1 (1)
Also these tangents intersect, then point of intersection is
t11(x+at12)=t21(x+at22)
from equation (1), I can write,
(x+at12)=−t12(x+at121)
(1−t12)x=−a(1−t12)
⟹x=−a
It is equation of the directrix.
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