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Sagot :
The linear equation of the tangent at point A is given by:
[tex]y = \frac{-4x + 25}{3}[/tex]
What is a linear function?
A linear function is modeled by:
y = mx + b
In which:
- m is the slope, which is the rate of change, that is, by how much y changes when x changes by 1.
- b is the y-intercept, which is the value of y when x = 0, and can also be interpreted as the initial value of the function.
In this problem, to find the slope, we need to find the derivative at (4,3), using implicit differentiation. Hence:
[tex]2x\frac{dx}{dx} + 2y\frac{dy}{dx} = 0[/tex]
[tex]\frac{dy}{dx} = -\frac{x}{y}[/tex]
[tex]\frac{dy}{dx} = -\frac{4}{3}[/tex]
Then:
[tex]y = -\frac{4}{3}x + b[/tex]
We use point (4,3) to find b, hence:
[tex]y = -\frac{4}{3}x + b[/tex]
[tex]3 = -\frac{16}{3} + b[/tex]
[tex]b = \frac{25}{3}[/tex]
Hence, the equation is:
[tex]y = \frac{-4x + 25}{3}[/tex]
More can be learned about linear equations at https://brainly.com/question/24808124
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