Consider the curve given by the equation y^(2)-6y-9x^(2)-144 x=576". It " can be shown that (dy)/(dx)=(9(x+8))/(y-3)". " Find the x-coordinate of the point where the line tangent to the curve is the x-axis. x=

Understand the problem: Understand the problem. A horizontal tangent line means the slope of the tangent line is 0. We are given the derivative of y with respect to x, (dy/dx), which represents the slope of the tangent line at any point on the curve. We need to find the x-coordinate where this slope is 0.Set derivative equal: Set the derivative equal to zero to find the x-coordinate. (dy/dx) = (9(x + 8)) / (y - 3) = 0 For the fraction to be zero, the numerator must be zero (since the denominator cannot be zero as it would make the expression undefined). 9(x + 8) = 0Solve for x: Solve for x. 9(x + 8) = 0 x + 8 = 0 x = -8

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Consider the curve given by the equation

y^(2)-6y-9x^(2)-144 x=576". It "
can be shown that

(dy)/(dx)=(9(x+8))/(y-3)". "
Find the
x-coordinate of the point where the line tangent to the curve is the
x-axis.

x=

Consider the curve given by the equation $y^{2}-6 y-9 x^{2}-144 x=576 \text {. }$ It can be shown that $\frac{d y}{d x}=\frac{9(x+8)}{y-3} \text {. }$ $\newline$ Find the $x$ -coordinate of the point where the line tangent to the curve is the $x$ -axis. $\newline$ $x=$

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Algebra 1

Transformations of quadratic functions

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Q. Consider the curve given by the equation

y^{2}-6 y-9 x^{2}-144 x=576 \text {. }

It can be shown that

\frac{d y}{d x}=\frac{9(x+8)}{y-3} \text {. }

\newline

Find the

x

-coordinate of the point where the line tangent to the curve is the

x

-axis.

\newline

x=

Understand the problem: Understand the problem. $\newline$ A horizontal tangent line means the slope of the tangent line is $0$ . We are given the derivative of $y$ with respect to $x$ , $\frac{dy}{dx}$ , which represents the slope of the tangent line at any point on the curve. We need to find the $x$ -coordinate where this slope is $0$ .
Set derivative equal: Set the derivative equal to zero to find the $x$ -coordinate. $(\frac{dy}{dx}) = \frac{9(x + 8)}{(y - 3)} = 0$ For the fraction to be zero, the numerator must be zero (since the denominator cannot be zero as it would make the expression undefined). $9(x + 8) = 0$
Solve for x: Solve for x. $\newline$ $9(x + 8) = 0$ $\newline$ $x + 8 = 0$ $\newline$ $x = -8$