## Application of Derivatives

# Tangents and Normals

- The equation of the tangent at (x
_{0}, y_{0}) 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 (x
_{0}, y_{0}), then the tangent at this point is parallel to the y-axis and its equation is x = x_{0}. - If tangent to a curve y = f(x) at x = x
_{0}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 (x
_{0}, y_{0}) 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 (x
_{0}, y_{0}) is zero, then equation of the normal is x = x_{0}. - If \frac{dy}{dx} at the point (x
_{0}, y_{0}) does not exist, then the normal is parallel to x-axis and its equation is y = y_{0}.

**Tangents and Normals - Tips**

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

**Tricks on tangents and Normals:**

- The tangent at any point of the curve x = at
^{3}, y = at^{4}divides abscissa of point of contact in the ratio 1 : 3. - Point on the curve ay
^{2}= x^{3}, 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 x
^{2}+ y^{2}= a^{2}is c^{2}= a^{2}(1 + m^{2}) - The condition y = mx + c is a tangent to the parabola y
^{2}= 4ax is c=\frac{a}{m} - The condition y = mx + c is a tangent to the parabola x
^{2}= 4ay is c = − a m^{2}. - 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 c
^{2}= a^{2}m^{2}+ b^{2}. - 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 c
^{2}= a^{2}m^{2}− b^{2}. - The curves y
^{2}= 4ax and x^{2}= 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 = c
^{2}, y^{2 }= 4ax cut each other orthogonally then c^{4}= 32 a^{4}. - The condition the two curves x = y
^{2}, xy = k cut at right angles iff 8k^{2}= 1. - Angle between the curves y
^{2}= 4ax and x^{2}= 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 a
_{1}x^{2}+ b_{1}y^{2}= 1, a_{2}x^{2}+ b_{2}y^{2}= 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(x
_{1}, y_{1}) 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(x
_{1}, y_{1}) 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(x
_{1}, y_{1}) 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(x
_{1}, y_{1}) to the curve with coordinate axes is \frac{(x_1+my_1)^{2}}{2|m|}

### View the Topic in this video From 00:40 To 34:54

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1. Equation of the tangent to the curve at P is Y-y=\frac{dy}{dx}(X-x)

2. The equation of normal at (x, y) to the curve is Y-y=-\frac{dx}{dy}(X-x)

3. **Slope of Tangent:**

(i) If the tangent at P is perpendicular to x-axis or parallel to y-axis, then \theta=\frac{\pi}{2} ⇒ tan θ = ∞ ⇒ \left(\frac{dy}{dx}\right)_{P}=\infty.

(ii) If the tangent at P is perpendicular to y-axis or parallel to x-axis, then θ = 0 ⇒ tan θ = 0 ⇒ \left(\frac{dy}{dx}\right)_{P}=0.

4. **Slope of Normal:**

(i) Slope of the normal at P=\frac{-1}{Slope \ of \ the \ tangent \ at \ P}

=\frac{-1}{\left(\frac{dy}{dx}\right)_{P}}=-\left(\frac{dx}{dy}\right)_{P}

(ii) If \left(\frac{dy}{dx}\right)_{P}=0, then normal at (x, y) is parallel to y-axis and perpendicular to x-axis.

(iii) If \left(\frac{dy}{dx}\right)_{P}=\infty, then normal at (x, y) is parallel to x-axis and perpendicular to y-axis.

5. **Length of Tangent and Normal:**

(i) Length of tangent, y \ cosec \ \theta=\frac{y\sqrt{1+\left(\frac{dy}{dx}\right)^{2}}}{\left(\frac{dy}{dx}\right)}

(ii) Length of normal, y \sec \ \theta=y{\sqrt{1+\left(\frac{dy}{dx}\right)^{2}}}

(iii) Length of sub tangent, y \cot \ \theta=\frac{y}{{\left(\frac{dy}{dx}\right)}}

(iv) Length of sub normal, y \tan \ \theta={y}{{\left(\frac{dy}{dx}\right)}}