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Equation of Normal to Parabola

Since the equation of the tangent to the parabola y2 = 4ax at (x1, y1) is
 
yy1 = 2a (x + x1) ...(7)
 
70772.png
 
The slope of the tangent at (x1, y1) is 2a/y1. Now the normal at (x1, y1) is perpendicular to the tangent at (x1, y1). Therefore, the slope of normal at (x1, y1) is –y1/2a. Hence the equation of normal at (x1, y1) is
 
yy1 = 70766.png (xx1)  ...(8)

Normal in parametric form

Replacing x1 by at2 and y1 by 2at in (8), we have y – 2at = – t (xat2) or y = – tx + 2at + at3.

Normal in slope form

The equation of normal to the parabola y2 = 4ax at (at2, 2at) is y = – tx + 2at + at3. Since m is the slope of the normal then m = –t. Then equation of normal is
 
y = mx – 2amam3 ...(9)
 
Thus y = mx –2amam3 is a normal to the parabola y2 = 4ax where m is the slope of the normal.
 
The coordinates of the foot of normal are (am2–2am). Comparing (9) with y = mx + c we have c = –2amam3, which is condition if y = mx + c is the normal of y2 = 4ax.

Properties of normal

  1. Normal to the parabola other than axis never passes through focus or focal chord can never be a normal chord.
  2. Point of intersection of normals at points
     
    P ≡ (at12, 2at1) and Q ≡ (at22, 2at2) to the parabola is
     
    R ≡ [2a + a(t12 + t22 + t1t2), – at1t2 (t1 + t2)]
  3. Normal at point P(at12, 2at1), y = – t1x + 2at1 + at13 meet the parabola again at point Q (at22, 2at2) such that t2 = 70760.png
  4. Normal at point P(t) meets the parabola again at point Q at angle tan–170754.png.

Reflection property of parabola

Any ray sent parallel to the axis of parabola passes through the focus of the parabola after reflected by the parabola.




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