Theorem1 the equation of tangent to the parabola y2 = 4ax at the point (x1, y1) is yy1 = 2a(x + x1) and the equation of normal at this point is 2a(y – y1) + y(x – x1) = 0
Proof: let P(x1, y1) and Q(x2, y2) lie on the parabola y2 = 4ax, also we have y12 = 4ax1, y22 = 4ax2
The equation of chord
y – y1 ‗ y2 – y1
x – x1 x2 – x1 …..(i)
The point P and Q lie on the parabola y2 = 4ax we have y12 = 4ax1 and y22 = 4ax2
y2 – y1 4a
x2 – x1 y1 + y2 ….(ii)
using (i) and (ii) we get the equation of chord PQ as
y – y1 4a
x – x1 y1 + y2
When Q→ P , then x2 → x1 ,y2 → y1 and chord PQ tends to tangent P. hence the equation of tangent at P is given by
y – y1 2a
x – x1 y1
yy1 = 2a(x – x1) + y12 = 2a(x – x1) + 4ax1
yy1 = 2a(x + x1)
this is the required equation of the tangent to the parabola.
The slope of the tangent is given by; 2a
y1
Slope of the normal is given by; -y1
2a
Hence the equation of the normal
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