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A curve $C$ is defined by the parametric equations $x(t) = 10\cos(t)$ and $y(t) = -2\cos(10t)$ . Write the equation of the line tangent to $C$ when $t = \frac{7\pi}{6}$ .

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Convert equations of conic sections from general to standard form

Full solution

Q. A curve

C

is defined by the parametric equations

x(t) = 10\cos(t)

and

y(t) = -2\cos(10t)

. Write the equation of the line tangent to

C

when

t = \frac{7\pi}{6}

.

Find Derivatives: First, we need to find the derivatives of $x(t)$ and $y(t)$ to get the slopes of the tangent lines.
Substitute $t$ Value: Next, we substitute $t = \frac{7\pi}{6}$ into the derivatives to find the slope of the tangent line at that point.
Calculate Tangent Slope: Calculate the slope of the tangent line using $\frac{dy}{dt}$ divided by $\frac{dx}{dt}.$
Find Coordinates: Find the coordinates of the point on the curve at $t = \frac{7\pi}{6}$ to use in the point-slope form of the line equation.
Use Point-Slope Form: Use the point-slope form of the line equation, $y - y_1 = m(x - x_1)$ , to write the equation of the tangent line.

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