Question

Find the equation of the tangent and the normal to the following curve at the indicated points x = at², y = 2at at t = 1 ?

Answer 1

\[x = a t^2 \text { and } y = 2at\]

\[\frac{dx}{dt} = 2at \text { and } \frac{dy}{dt} = 2a\]

\[ \therefore \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{2a}{2at} = \frac{1}{t}\]

\[\text { Slope of tangent },m= \left( \frac{dy}{dx} \right)_{t = 1} =\frac{1}{1}=1\]

\[\text { Now }, \left( x_1 , y_1 \right) = \left( a, 2a \right)\]

\[\text { Equation of tangent is },\]

\[y - y_1 = m \left( x - x_1 \right)\]

\[ \Rightarrow y - 2a = 1\left( x - a \right)\]

\[ \Rightarrow y - 2a = x - a\]

\[ \Rightarrow x - y + a = 0\]

Equation of normal:

\[y - y_1 = m \left( x - x_1 \right)\]

\[ \Rightarrow y - 2a = - 1 \left( x - a \right)\]

\[ \Rightarrow y - 2a = - x + a\]

\[ \Rightarrow x + y = 3a\]