MATH 121 (Wg Cdr Monir)
Tangent and Normal
1. The tangent at P to a given curve is defined as the limiting position of the secant PQ (when such a limit exists) as the point Q approaches P along the curve.
Derivative of any point on a curve is the slope of the tangent at the point (x,y) on the curve.
That is \(\frac{dy}{dx} = \tan \psi = \) slope of the tangent.
In a curve \(y = f(x)\), if at a particular point (x,y)
1. \(\frac{dy}{dx} = 1\), Then the slope of the tangent at that point of the curve makes an angle of \(45^\circ\) with the positive direction of x-axis.
2. \(\frac{dy}{dx}\) = Negative, Then the tangent to the curve at that point makes an obtuse angle with the positive direction of x-axis.
3. \(\frac{dy}{dx} = 0\) , Then the tangent to the curve at that point makes an angle of \(0^\circ\) with the x-axis, i.e the tangent to the curve at that point is parallel to the x-axis.
4. \(\frac{dy}{dx} = \infty\), then the tangent to the curve at that point makes on angle of \(\frac{\pi}{2}\) with the x-axis, i.e the tangent to the curve at the point is perpendicular to x-axis.
Equation of tangent at any point (x,y) of the curve \(y = f(x)\)
$$ Y - y = \frac{dy}{dx} (X - x) $$
Equation of the tangent at any point on the curve \(f(x,y) = 0\) $$ (X - x) f_x + (Y - y) f_y = 0 $$
If the curve be given by the algebraic equation \(f(x,y) = 0\) which can be made homoheneous in x,y and z the equation of the tangent will be \(X f_x + Y f_y + Z f_x = 0\)
Equation of the tangent at any point on the curve \(f(x,y) = 0\)
if the equation of the curve be given in the form \(f(x,y) = 0\) then
\(\frac{dy}{dx} = \frac{ - \frac{\partial f}{\partial x}}{\frac{\partial f }{\partial y}}\)
substituting this expression for \(\frac{dy}{dx}\) in
\begin{align*}
& Y - y = \frac{dy}{dx} (X - x)\\
\textrm{we have} &, \\
& Y - y = \frac{ - \frac{\partial f}{\partial x}}{\frac{\partial f }{\partial y}} (X - x)\\
or, ~ & (X - x) \frac{\partial f}{\partial x} + (Y - y) \frac{ \partial f}{\partial y} = 0
\end{align*}
as the equation of the tangent.
Replacing \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\) by \(f_x\) and \(f_y\) respectively we may write the equation of the tangent in the form
\((X - x) f_x + (Y - y) f_y = 0\)
♦ Normal and its equation.
Normal: Normal to a curve at a point is defined as the straight line through that point at right angles to the tangent to the curve at that point.
We know that tangent to a curve \( y = f(x) ~ \textrm{at} ~ (x,y) ~\textrm{is} \)
\(Y - y = \frac{dy}{dx} (X - x) \tag{1}\label{eq:1}\)
any line throught(x,y) is given by
\(Y - y = m (X - x) \tag{2}\label{eq:2}\)
If this line is normal to the curve at (x,y) this is perpendicular to the tangent to the curve, i.e
\begin{align*}
& m \frac{dy}{dx} = - 1\\
\therefore ~& m = - \frac{1}{\frac{dy}{dx}}
\end{align*}
From (2) the equation of the normal at (x,) to the curve \(y = f(x)\) is
\begin{align*}
& Y - y = - \frac{1}{\frac{dy}{dx}} (X - x) \\
or, ~ & (Y - y ) \frac{dy}{dx} + (X - x) = 0
\end{align*}
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