Find an equation of the tangent line to the graph of the function $( y - 6 ) ^ { 2 } = 5 ( x - 5 )$ at the point $( 8.20,2.00 )$. The coefficients below are given to two decimal places.
A) $y = - 0.63 x + 7.13$
B) $y = 3.38 x + 25.68$
C) $y = 3.38 x - 25.68$
D) $y = - 0.63 x + 25.68$
E) $y = 0.63 x + 7.13$

Question

Quizplus · Accepted Answer

To find the equation of the tangent line, we first need to differentiate the given equation implicitly to find the slope of the tangent line at the given point. The given equation is $(y - 6)^2 = 5(x - 5)$. Differentiating both sides with respect to $x$, we get $2(y - 6) \frac{dy}{dx} = 5$. At the point $(8.20, 2.00)$, we substitute $y = 2.00$ to find $\frac{dy}{dx}$, which gives us $2(2 - 6) \frac{dy}{dx} = 5$, leading to $\frac{dy}{dx} = -\frac{5}{8}$. The slope of the tangent line is $-\frac{5}{8}$, which approximates to $-0.63$. Using the point-slope form of a line, $y - y_1 = m(x - x_1)$, with $m = -0.63$ and the point $(8.20, 2.00)$, we find the equation of the tangent line, which simplifies to $y = -0.63x + 7.13$.

Find an Equation of the Tangent Line to the Graph (y−6)2=5(x−5)( y - 6 ) ^ { 2 } = 5 ( x - 5 )(y−6)2=5(x−5)

Find an Equation of the Tangent Line to the Graph $( y - 6 ) ^ { 2 } = 5 ( x - 5 )$