1. verify that the point is on the given curve. evaluate 4(x² + y²)² for x = 1 and y = 2. 2. determine an…

Explanation:

Step1: Substitute x and y values

Substitute (x = 1) and (y=2) into (4(x^{2}+y^{2})^{2}). First, calculate (x^{2}+y^{2}): [x^{2}+y^{2}=1^{2}+2^{2}=1 + 4=5]

Step2: Calculate the square - squared value

Then calculate ((x^{2}+y^{2})^{2}) and multiply by 4. Since ((x^{2}+y^{2}) = 5), then ((x^{2}+y^{2})^{2}=5^{2}=25), and (4(x^{2}+y^{2})^{2}=4\times25 = 100). Next, for the right - hand side of the equation (25xy^{2}), substitute (x = 1) and (y = 2): [25xy^{2}=25\times1\times2^{2}=25\times4=100] Since the left - hand side (4(x^{2}+y^{2})^{2}=100) and the right - hand side (25xy^{2}=100) when (x = 1) and (y = 2), the point ((1,2)) is on the curve.

Now, to find the equation of the tangent line, we use implicit differentiation. Differentiate (4(x^{2}+y^{2})^{2}=25xy^{2}) with respect to (x). Let (u=x^{2}+y^{2}), then the left - hand side is (4u^{2}). By the chain rule, (\frac{d}{dx}(4u^{2})=8u\frac{du}{dx}=8(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 = 25x) and (v=y^{2}), we have (\frac{d}{dx}(25xy^{2})=25y^{2}+50xy\frac{dy}{dx}). So, (8(x^{2}+y^{2})(2x + 2y\frac{dy}{dx})=25y^{2}+50xy\frac{dy}{dx}). Substitute (x = 1) and (y = 2): [8(1^{2}+2^{2})(2\times1+2\times2\frac{dy}{dx})=25\times2^{2}+50\times1\times2\frac{dy}{dx}] [8\times5(2 + 4\frac{dy}{dx})=100 + 100\frac{dy}{dx}] [40(2 + 4\frac{dy}{dx})=100 + 100\frac{dy}{dx}] [80+160\frac{dy}{dx}=100 + 100\frac{dy}{dx}] [160\frac{dy}{dx}-100\frac{dy}{dx}=100 - 80] [60\frac{dy}{dx}=20] [\frac{dy}{dx}=\frac{1}{3}] The equation of the tangent line using the point - slope form (y - y_{1}=m(x - x_{1})) with ((x_{1},y_{1})=(1,2)) and (m=\frac{1}{3}) is: [y - 2=\frac{1}{3}(x - 1)] [y=\frac{1}{3}x-\frac{1}{3}+2] [y=\frac{1}{3}x+\frac{5}{3}]

Answer:

The point ((1,2)) is on the curve (4(x^{2}+y^{2})^{2}=25xy^{2}). The equation of the tangent line to the curve at the point ((1,2)) is (y=\frac{1}{3}x+\frac{5}{3})