Tangents and Normals

• The equation of the tangent at (x₀, y₀) to the curve y = f(x) is given by
  y-y_0=\frac{dy}{dx}]_{x-x_0}\ (x-x_0)
• If \frac{dy}{dx} does not exist at the point (x₀, y₀), then the tangent at this point is parallel to the y-axis and its equation is x = x₀.
• If tangent to a curve y = f(x) at x = x₀ is parallel to x-axis, then \frac{dy}{dx}]_{x=x_0}=0.
• Equation of the normal to the curve y = f(x) at a point (x₀, y₀) is given by
  y-y_0=\frac{-1}{\frac{dy}{dx}]_{(x_0,y_0)}}(x-x_0)
• If \frac{dy}{dx} at the point (x₀, y₀) is zero, then equation of the normal is x = x₀.
• If \frac{dy}{dx} at the point (x₀, y₀) does not exist, then the normal is parallel to x-axis and its equation is y = y₀.

Tangents and Normals - Tips

• Slope of tangent to the curve y = f(x) at P(x₁, y₁) is m = f'(x₁) and equation of tangent is y  − y₁ = m (x − x₁)
• Slope of Normal to the curve y = f(x) at P(x₁, y₁) is \frac{-1}{m}=\frac{-1}{f'(x_1)} and equation of normal is y-y_1=\frac{-1}{m}(x-x_1)
• Length of tangent at P(x₁, y₁) is \begin{vmatrix}\frac{y_{1}\sqrt{1+m^{2}}}{m}\end{vmatrix}
• Length of sub tangent at P(x₁, y₁) is \begin{vmatrix}\frac{y_{1}}{m}\end{vmatrix}
• Length of Normal at P(x₁, y₁) is \begin{vmatrix}y_{1}\sqrt{1+m^{2}}\end{vmatrix}
• Length of sub normal at P(x₁, y₁) is |y_pm|

Tricks on tangents and Normals:

• The tangent at any point of the curve x = at³, y = at⁴ divides abscissa of point of contact in the ratio 1 : 3.
• Point on the curve ay² = x³, the normal at which makes equal intercepts on the axes is \left(\frac{4a}{9},\frac{8a}{27}\right)
• The condition y = mx + c is a tangent to the circle x² + y² = a² is c² = a²(1 + m²)
• The condition y = mx + c is a tangent to the parabola y² = 4ax is c=\frac{a}{m}
• The condition y = mx + c is a tangent to the parabola x² = 4ay is c = − a m².
• The condition that the line y = mx + c is tangent to the ellipse \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 is c² = a²m² + b².
• The condition that the line y = mx + c is tangent to the hyperbola \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 is c² = a²m² − b².
• The curves y² = 4ax and x² = 4ay intersect at (0, 0), (4a, 4a). At (0, 0) the curves are orthogonal and (4a, 4a) the angle between the curves is \tan^{-1}\left(\frac{3}{4}\right)
• If the curves xy = c², y² = 4ax cut each other orthogonally then c⁴ = 32 a⁴.
• The condition the two curves x = y², xy = k cut at right angles iff 8k² = 1.
• Angle between the curves y² = 4ax and x² = 4by at their point of intersection is \theta=\tan^{-1}\left(\frac{3a^{1/3}b^{1/3}}{2(a^{2/3}+b^{2/3})}\right)
• If the curves a₁x² + b₁y² = 1, a₂x² + b₂y² = 1 cut each other orthogonally, then \frac{1}{a_{1}}-\frac{1}{a_{2}}=\frac{1}{b_{1}}-\frac{1}{b_{2}}
• The area of the triangle formed by the tangent and normal at p(x₁, y₁) and x-axis is \frac{y_1^2(1+m^{2})}{2|m|} sq.units.
• The area of the triangle formed by the tangent and normal at p(x₁, y₁) and y-axis is \frac{x_1^2(1+m^{2})}{2|m|} sq.units.
• The area of the triangle formed by the tangent at p(x₁, y₁) to the curve with coordinate axes is \frac{(y_1-mx_1)^{2}}{2|m|}
• The area of the triangle formed by the normal at p(x₁, y₁) to the curve with coordinate axes is \frac{(x_1+my_1)^{2}}{2|m|}