let us consider the point P(a,2a) on the parabola
The rough diagram of the question is attached here...
from P the normal to the parabola is drawn which intersects the x axis at the point Q,the coordinate of which is (3a,0). ( working of Q is shown below).
from Q a line perpendicular to PQ is drawn and we have to show that it intersects the parabola
Now equation of tangent at the point P(a,2a) is given by
(Here slope (m) of tangent is 1.)
Now Equation of Normal (which is perpendicular to the tangent at P ) is given by
AS this normal intersects X-axis at Q. so ordinate of Q must be '0'.
putting y=0 in
we get x=3a.
The coordinate of Q is (3a,0).
Now slope of the line which is passing through Q (3a,0) and perpendicular to the normal at P
is again as product of slope of two perpendiculars line is always .
The equation of line passing through Q(3a,0) is given by
Now the coordinates of the end points of the latus rectum of the parabola
clearly the Coordinate (a,-2a) satisfies both
Hence the required line intersects the parabola
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