a. verify that the given point lies on the curve. b. determine an equation of the line tangent to the curve…

Explanation:

Step1: Substitute x and y values

Substitute (x = 3) and (y=4) into the left - hand side of the equation (48(x^{2}+y^{2})^{2}). [ \begin{align*} x^{2}+y^{2}&=3^{2}+4^{2}\ &=9 + 16\ &=25 \end{align*} ] Then ((x^{2}+y^{2})^{2}=25^{2}=625), and (48(x^{2}+y^{2})^{2}=48\times625 = 30000).

Step2: Substitute x and y values into the right - hand side

Substitute (x = 3) and (y = 4) into the right - hand side of the equation (625xy^{2}). [ \begin{align*} 625xy^{2}&=625\times3\times4^{2}\ &=625\times3\times16\ &=625\times48\ &=30000 \end{align*} ] Since the left - hand side equals the right - hand side when (x = 3) and (y = 4), the point ((3,4)) lies on the curve.

Step3: Differentiate the equation implicitly

Differentiate (48(x^{2}+y^{2})^{2}=625xy^{2}) with respect to (x). Let (u=x^{2}+y^{2}), then (\frac{d}{dx}[48(x^{2}+y^{2})^{2}]=48\times2(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 = 625x) and (v=y^{2}), we have (\frac{d}{dx}(625xy^{2})=625y^{2}+625x\times2y\frac{dy}{dx}). So (96(x^{2}+y^{2})(2x + 2y\frac{dy}{dx})=625y^{2}+1250xy\frac{dy}{dx}). Substitute (x = 3) and (y = 4) into the above equation: First, (x^{2}+y^{2}=3^{2}+4^{2}=25). [ \begin{align*} 96\times25\times(2\times3 + 2\times4\frac{dy}{dx})&=625\times4^{2}+1250\times3\times4\frac{dy}{dx}\ 96\times25\times(6 + 8\frac{dy}{dx})&=625\times16+15000\frac{dy}{dx}\ 14400+19200\frac{dy}{dx}&=10000 + 15000\frac{dy}{dx}\ 19200\frac{dy}{dx}-15000\frac{dy}{dx}&=10000 - 14400\ 4200\frac{dy}{dx}&=- 4400\ \frac{dy}{dx}&=-\frac{4400}{4200}=-\frac{22}{21} \end{align*} ] The equation of the tangent line using the point - slope form (y - y_{1}=m(x - x_{1})) where ((x_{1},y_{1})=(3,4)) and (m =-\frac{22}{21}) is (y - 4=-\frac{22}{21}(x - 3)), which simplifies to (21y-84=-22x + 66), or (22x+21y=150).

Answer:

a. The point ((3,4)) lies on the curve since when (x = 3) and (y = 4), (48(x^{2}+y^{2})^{2}=625xy^{2}=30000). b. The equation of the tangent line to the curve at the point ((3,4)) is (22x + 21y=150).