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Equation of Tangents and Normals

Let P(x1, y1) be any point on the curve y = f(x).
 
91421.png
 
If tangent at P makes angle θ with positive direction of the x-axis, then dy/dx = tan θ.
 
Equation of tangent Equation of tangent at point P(x1, y1) is
yy1 = 91409.png
 
Equation of normal Equation of normal at point P(x, y) is
yy1 = 91402.png
 
Notes:
  • The point P(x1y1) will satisfy the equation of the curve and the equation of tangent and normal line.
  • If the tangent at any point P on the curve is parallel to the axis of x then dy/dx = 0 at the point P.
  • If the tangent at any point on the curve is parallel to the axis of y, then dy/dx = ∞ or dx/dy = 0.
  • If the tangent at any point on the curve is equally inclined to both the axes then dy/dx = ±1.
  • If the tangent at any point makes equal intercept on the coordinate axes then dy/dx = –1.
  • Tangent to a curve at the point P(x1y1) can be drawn even though dy/dx at P does not exist.
     
    For example, x = 0 is a tangent to y = x2/3 at (0, 0).




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