Question

a. verify that the given point lies on the curve. b. determine an equation of the line tangent to the curve at the given point. 2x² + 2xy + 3y² = 18; (1,2) a. verify that the point is on the given curve. it is given that the right side of the equation equals 18. evaluate the left side, 2x² + 2xy + 3y², when x is 1 and y is 2. when x is 1 and y is 2, 2x² + 2xy + 3y² = (simplify your answer.)

Explanation:

Step1: Evaluate left - hand side for part a

Substitute \(x = 1\) and \(y=2\) into \(2x^{2}+2xy + 3y^{2}\).
\[

$$\begin{align*} 2x^{2}+2xy+3y^{2}&=2\times(1)^{2}+2\times1\times2 + 3\times(2)^{2}\\ &=2\times1+4 + 3\times4\\ &=2 + 4+12\\ &=18 \end{align*}$$

\]

Step2: Differentiate the equation implicitly for part b

Differentiate \(2x^{2}+2xy + 3y^{2}=18\) with respect to \(x\).
Using the sum - rule and product - rule:
The derivative of \(2x^{2}\) is \(4x\).
For \(2xy\), by the product - rule \((uv)^\prime=u^\prime v+uv^\prime\) where \(u = 2x\) and \(v = y\), we get \(2y+2x\frac{dy}{dx}\).
For \(3y^{2}\), using the chain - rule, we get \(6y\frac{dy}{dx}\).
So, \(4x + 2y+2x\frac{dy}{dx}+6y\frac{dy}{dx}=0\).

Step3: Solve for \(\frac{dy}{dx}\)

Group the terms with \(\frac{dy}{dx}\) together:
\(2x\frac{dy}{dx}+6y\frac{dy}{dx}=-4x - 2y\).
Factor out \(\frac{dy}{dx}\): \(\frac{dy}{dx}(2x + 6y)=-4x - 2y\).
Then \(\frac{dy}{dx}=\frac{-4x - 2y}{2x+6y}=\frac{-2x - y}{x + 3y}\).

Step4: Find the slope of the tangent line

Substitute \(x = 1\) and \(y = 2\) into \(\frac{dy}{dx}\):
\(\frac{dy}{dx}\big|_{(1,2)}=\frac{-2\times1-2}{1 + 3\times2}=\frac{-2 - 2}{1+6}=\frac{-4}{7}\).

Step5: Find the equation of the tangent line

Use the point - slope form \(y - y_{1}=m(x - x_{1})\), where \((x_{1},y_{1})=(1,2)\) and \(m =-\frac{4}{7}\).
\(y - 2=-\frac{4}{7}(x - 1)\).
Multiply through by 7 to get \(7y-14=-4x + 4\).
Rearrange to \(4x+7y=18\).

Answer:

a. Since substituting \(x = 1\) and \(y = 2\) into \(2x^{2}+2xy + 3y^{2}\) gives \(18\), the point \((1,2)\) lies on the curve.
b. The equation of the tangent line is \(4x + 7y=18\).