Tangents and Normal form ppt

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tangent and normal form ppt

Any line that touch any curve or slope in any side is called tangent, in other words, A tangent to a curve is a line that touch the curve without cutting through it. Tangents and Normal form ppt.

We know that for any curve whose equation is given by y= f(x)    or     f(x, y) = 0   the derivation d y / d x is the slope of the tangent at any point P(x, y) to the curve. The equation of the tangent to the curve can easily be written by the point slope formula .

Any line that perpendicular to the tangent line at any point or any side is called a normal line. In other words,  the normal line to the curve at P is the line through P perpendicular to the tangent to the curve  at P. this method can be very conventicle employed  to fined of equation of tangent and normal form ppt to the circle.

Tangents and Normal form ppt image.

tangent and normal form ppt

equation of tangent and normal to the circle

at the point at the point

Here

f(x, y) =equation of tangent and normal to the circle ……………….. A

differentiating (1) w, r,  t, x, we get

2x +2 y d y/d x+2g + 2f d y / d x =0

\frac{dy}{dx}= - \frac{x+g}{y+f}

if slope of tangent at point at the point

\frac{dy}{dx}]_{x_{1},y_{1}}= \frac{x_{1}+g}{y_{1}+f}

Hence, the equation of tangent  at point p is  with  point slope form

{y-y_{1}}= -\frac{x_{1}+g}{y_{1}+f}\left ( x-x_{1} \right )

algebraic simplification

y(y_{1}+f)-y_{1}^{2}-y_{1}f=-x(x_{1}+g)+x_{1}2+x_{1}g

xx_{1}+yy_{1}+gx+fy=x_{1}^{2}+y_{1}^{2}+gx_{1}+fy_{1}

adding both side    gx_{1}+fy_{1}+c  after implication

xx_{1}+yy_{1}+g(x+x_{1})+f\left ( y+y_{1} \right )+c=0

since  \left ( x_{1}, y_{1} \right ) lies on A and so on

x_{1}^{2}+y_{1}^{2}+2gx_{1}+2fy_{1}+c=0

xx_{1}+yy_{1}+g(x+x_{1})+f\left ( y+y_{1} \right )+c=0

Is the required equation of the tangent form ppt.

Equation of normal at P 

we fined an equation of the normal at P. We take the negative reciprocal of the slope of the tangent.

\frac{y_{1}+f}{x_{1}+g}

equation of normal at point at the point is

y-y_{1}= \frac{y_{1}+f}{x_{1}+g} \left ( x-x_{1} \right )

thus the equation of the normal line is

\left ( y-y_{1} \right )\left ( x_{1}+g \right )=\left ( x-x_{1} \right )\left ( y_{1}+ f\right )

required equation of normal form ppt

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