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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 ?
Solution
\[x = a t^2 \text { and } y = 2at\]
\[\frac{dx}{dt} = 2at \text { and } \frac{dy}{dt} = 2a\]
\[ \therefore \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{2a}{2at} = \frac{1}{t}\]
\[\text { Slope of tangent },m= \left( \frac{dy}{dx} \right)_{t = 1} =\frac{1}{1}=1\]
\[\text { Now }, \left( x_1 , y_1 \right) = \left( a, 2a \right)\]
\[\text { Equation of tangent is },\]
\[y - y_1 = m \left( x - x_1 \right)\]
\[ \Rightarrow y - 2a = 1\left( x - a \right)\]
\[ \Rightarrow y - 2a = x - a\]
\[ \Rightarrow x - y + a = 0\]
Equation of normal:
\[y - y_1 = m \left( x - x_1 \right)\]
\[ \Rightarrow y - 2a = - 1 \left( x - a \right)\]
\[ \Rightarrow y - 2a = - x + a\]
\[ \Rightarrow x + y = 3a\]
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