Respuesta :
Answer:
The equation of the tangent line of the given curve
[tex]\frac{dy}{dx} = \frac{- (2x +2y +1)}{( 2 x - 2 y)}[/tex]
The tangent of the given curve at the point
[tex](\frac{dy}{dx})_{(7,9)} = \frac{33}{4}[/tex]
Step-by-step explanation:
Explanation :-
Step(i):-
Given equation of the parabola
x²+2xy−y²+x=101 ...(i)
apply derivative Formulas
a) [tex]\frac{dx^{n} }{dx} = n x ^{n-1}[/tex]
b) [tex]\frac{d U V }{dx} = U \frac{dV}{dx} + V \frac{dU}{dx}[/tex]
Step(ii):-
Differentiating equation (i) with respective to 'x' , we get
[tex]2 x + 2 ( x \frac{dy}{dx} + y) - 2 y \frac{dy}{dx} +1 = 0[/tex]
[tex]2 x + 2 x \frac{dy}{dx} +2 y - 2 y \frac{dy}{dx} +1 = 0[/tex]
on simplification , we get
[tex]( 2 x - 2 y) \frac{dy}{dx} = - (2x +2y +1)[/tex]
[tex]\frac{dy}{dx} = \frac{- (2x +2y +1)}{( 2 x - 2 y)}[/tex]
The tangent of the given curve at the point ( 7,9)
[tex](\frac{dy}{dx})_{(7,9)} = \frac{- ((2(7) +2(9) +1))}{( 2 (7) - 2 (9)}[/tex]
[tex](\frac{dy}{dx})_{(7,9)} = \frac{- (33)}{( -4} = \frac{33}{4}[/tex]
Final answer :-
The equation of the tangent line of the given curve
[tex]\frac{dy}{dx} = \frac{- (2x +2y +1)}{( 2 x - 2 y)}[/tex]
The tangent of the given curve at the point
[tex](\frac{dy}{dx})_{(7,9)} = \frac{33}{4}[/tex]
