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 dydx=tanψ=dydx=tanψ= slope of the tangent.
In a curve y=f(x)y=f(x), if at a particular point (x,y)
1. dydx=1dydx=1, Then the slope of the tangent at that point of the curve makes an angle of 45∘45∘ with the positive direction of x-axis.
2. dydxdydx = Negative, Then the tangent to the curve at that point makes an obtuse angle with the positive direction of x-axis.
3. dydx=0dydx=0 , Then the tangent to the curve at that point makes an angle of 0∘0∘ with the x-axis, i.e the tangent to the curve at that point is parallel to the x-axis.
4. dydx=∞dydx=∞, then the tangent to the curve at that point makes on angle of π2π2 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=f(x)
Y−y=dydx(X−x)Y−y=dydx(X−x)
Equation of the tangent at any point on the curve f(x,y)=0f(x,y)=0 (X−x)fx+(Y−y)fy=0(X−x)fx+(Y−y)fy=0
If the curve be given by the algebraic equation f(x,y)=0f(x,y)=0 which can be made homoheneous in x,y and z the equation of the tangent will be Xfx+Yfy+Zfx=0Xfx+Yfy+Zfx=0
Equation of the tangent at any point on the curve f(x,y)=0f(x,y)=0
if the equation of the curve be given in the form f(x,y)=0f(x,y)=0 then
dydx=−∂f∂x∂f∂ydydx=−∂f∂x∂f∂y
substituting this expression for dydxdydx in
Y−y=dydx(X−x)we have,Y−y=−∂f∂x∂f∂y(X−x)or, (X−x)∂f∂x+(Y−y)∂f∂y=0
as the equation of the tangent.
Replacing ∂f∂x and ∂f∂y by fx and fy respectively we may write the equation of the tangent in the form
(X−x)fx+(Y−y)fy=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) at (x,y) is
Y−y=dydx(X−x)
any line throught(x,y) is given by
Y−y=m(X−x)
If this line is normal to the curve at (x,y) this is perpendicular to the tangent to the curve, i.e
mdydx=−1∴ m=−1dydx
From (2) the equation of the normal at (x,) to the curve y=f(x) is
Y−y=−1dydx(X−x)or, (Y−y)dydx+(X−x)=0
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