verify that the given point is on the curve. find the lines that are (a) tangent and (b) normal to the curve…

verify that the given point is on the curve. find the lines that are (a) tangent and (b) normal to the curve at the given point. 6xy + π sin y = 61π, (4, 5π/2) statement is 61π = 61π, which is a true statement. (type an exact answer, using π as needed.) a. write the equation of the tangent line to the curve at (4, 5π/2). y = - 5π/8 x + 5π (type an equation. type an exact answer, using π as needed.) b. write the equation of the normal line to the curve at (4, 5π/2). (type an equation. type an exact answer, using π as needed.)

verify that the given point is on the curve. find the lines that are (a) tangent and (b) normal to the curve at the given point. 6xy + π sin y = 61π, (4, 5π/2) statement is 61π = 61π, which is a true statement. (type an exact answer, using π as needed.) a. write the equation of the tangent line to the curve at (4, 5π/2). y = - 5π/8 x + 5π (type an equation. type an exact answer, using π as needed.) b. write the equation of the normal line to the curve at (4, 5π/2). (type an equation. type an exact answer, using π as needed.)

Answer

Explanation:

Step1: Recall relationship between tangent and normal slopes

The slope of the normal line is the negative - reciprocal of the slope of the tangent line. The equation of the tangent line is $y =-\frac{5\pi}{8}x + 5\pi$, so its slope $m_t=-\frac{5\pi}{8}$.

Step2: Calculate slope of normal line

The slope of the normal line $m_n$ is the negative - reciprocal of $m_t$. So $m_n=\frac{8}{5\pi}$.

Step3: Use point - slope form to find equation of normal line

The point - slope form of a line is $y - y_1=m(x - x_1)$, where $(x_1,y_1)=(4,\frac{5\pi}{2})$ and $m = \frac{8}{5\pi}$. Substitute the values: $y-\frac{5\pi}{2}=\frac{8}{5\pi}(x - 4)$. Expand: $y-\frac{5\pi}{2}=\frac{8}{5\pi}x-\frac{32}{5\pi}$. Add $\frac{5\pi}{2}$ to both sides: $y=\frac{8}{5\pi}x-\frac{32}{5\pi}+\frac{5\pi}{2}$. Get a common denominator: $y=\frac{8}{5\pi}x+\frac{- 64 + 125\pi^{2}}{10\pi}$.

Answer:

$y=\frac{8}{5\pi}x+\frac{- 64 + 125\pi^{2}}{10\pi}$