Find an equation of the tangent line to the graph at the given point. ? (y - 3)² = 4(x - 5), (6, 5)

Solution:

Given, the equation is (y - 3)² = 4(x - 5)

We have to find the equation of the tangent line to the graph at the point (6, 5).

On expanding,

y² - 6y + 9 = 4x - 20

y² - 6y - 4x = -20 - 9

y² - 6y - 4x = -29

On differentiating,

2y(dy/dx) - 6(dy/dx) - 4 = 0

(dy/dx)[2y - 6] = 4

dy/dx = 4/(2y-6)

dy/dx = 2/(y - 3)

At the point (6, 5)

dy/dx = 2/(5 - 3)

dy/dx = 2/2

dy/dx = 1

The equation of the line in slope point form is given by

(y - y₁) = m(x - x₁)

(y - 5) = 1(x - 6)

y = x - 6 + 5

y = x - 1

Therefore, the equation of the tangent line is y = x -1.

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Find an equation of the tangent line to the graph at the given point. ? (y - 3)² = 4(x - 5), (6, 5)

Summary:

An equation of the tangent line to the graph (y - 3)² = 4(x - 5) at the given point (6, 5) is y = x - 1.