the graph of 18(x² + y²)² = 169xy², shown in the figure, is a double folium. find the equation of the…

Explanation:

Step1: Differentiate both sides implicitly

Differentiate $18(x^{2}+y^{2})^{2}=169xy^{2}$ with respect to $x$. Using the chain - rule and product - rule: Let $u = x^{2}+y^{2}$, then the left - hand side is $18u^{2}$. Its derivative is $18\times2u\times(2x + 2y\frac{dy}{dx})=36(x^{2}+y^{2})(2x + 2y\frac{dy}{dx})$. The right - hand side: Using the product rule $(uv)^\prime=u^\prime v+uv^\prime$ where $u = 169x$ and $v = y^{2}$. So its derivative is $169y^{2}+169x\times2y\frac{dy}{dx}$. So we have $36(x^{2}+y^{2})(2x + 2y\frac{dy}{dx})=169y^{2}+338xy\frac{dy}{dx}$.

Step2: Substitute the point $(2,3)$

Substitute $x = 2$ and $y = 3$ into the above equation: First, calculate $x^{2}+y^{2}=4 + 9=13$. The left - hand side is $36\times13\times(2\times2+2\times3\frac{dy}{dx})=468\times(4 + 6\frac{dy}{dx})=1872+2808\frac{dy}{dx}$. The right - hand side is $169\times9+338\times2\times3\frac{dy}{dx}=1521+2028\frac{dy}{dx}$. Then $1872+2808\frac{dy}{dx}=1521+2028\frac{dy}{dx}$.

Step3: Solve for $\frac{dy}{dx}$

Rearrange the equation $1872+2808\frac{dy}{dx}=1521+2028\frac{dy}{dx}$: $2808\frac{dy}{dx}-2028\frac{dy}{dx}=1521 - 1872$. $780\frac{dy}{dx}=- 351$. $\frac{dy}{dx}=-\frac{351}{780}=-\frac{9}{20}$.

Step4: Use the point - slope form

The point - slope form of a line is $y - y_{1}=m(x - x_{1})$, where $(x_{1},y_{1})=(2,3)$ and $m =-\frac{9}{20}$. $y - 3=-\frac{9}{20}(x - 2)$. $20y-60=-9x + 18$. $9x+20y=78$.

Answer:

$9x + 20y=78$