A-Level AQA Maths Mechanics Working with Vectors: A curve has equation $$x^2y^2 +

A curve has equation $$x^2y^2 + xy^4 = 12$$ 9 (a) Prove that the curve does not intersect the coordinate axes - AQA - A-Level Maths Mechanics - Question 9 - 2019 - Paper 3

Question 9

A curve has equation $$x^2y^2 + xy^4 = 12$$ 9 (a) Prove that the curve does not intersect the coordinate axes. 9 (b)(i) Show that $$\frac{dy}{dx} = \frac{2xy + y... show full transcript

Worked Solution & Example Answer:A curve has equation $$x^2y^2 + xy^4 = 12$$ 9 (a) Prove that the curve does not intersect the coordinate axes - AQA - A-Level Maths Mechanics - Question 9 - 2019 - Paper 3

Step 1

Prove that the curve does not intersect the coordinate axes.

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Answer

To show that the curve does not intersect the coordinate axes, we can set either $x = 0$ or $y = 0$ and analyze the implications.

Case 1: Setting $x = 0$

Substituting $x = 0$ into the equation: $0 + 0 = 12$ . This leads to a contradiction since $0 eq 12$ .
Case 2: Setting $y = 0$

Substituting $y = 0$ into the equation: $0 + 0 = 12$ . This also leads to a contradiction since $0 eq 12$ .

Since both cases result in contradictions, we conclude that the curve does not intersect either the x-axis or the y-axis.

Step 2

Show that $\frac{dy}{dx} = \frac{2xy + y^3}{2x^2 + 4xy^2}$.

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Answer

To find (\frac{dy}{dx}), we will differentiate the equation of the curve implicitly.

Differentiate the equation Using implicit differentiation on the equation $x^2y^2 + xy^4 = 12$ , we apply the product rule:
- The derivative of $x^2y^2$ :
  $\frac{d}{dx}(x^2y^2) = 2xy^2 + x^2\frac{dy}{dx} imes 2y$ .
- The derivative of $xy^4$ :
  $\frac{d}{dx}(xy^4) = y^4 + x(4y^3\frac{dy}{dx})$ .
- Thus, we have: $2xy^2 + (y^4 + 4xy^3\frac{dy}{dx}) = 0$ .
Collect terms involving (\frac{dy}{dx}) Rearranging the derivative terms: $4xy^3\frac{dy}{dx} = -2xy^2 - y^4$
Solving for (\frac{dy}{dx}) Factor out the terms in the equation: $\frac{dy}{dx} = \frac{-2xy^2 - y^4}{4xy^3}$ Simplifying gives: $\frac{dy}{dx} = \frac{2xy + y^3}{2x^2 + 4xy^2}.$ Thus, we have shown the required result.

A curve has equation $$x^2y^2 + xy^4 = 12$$ 9 (a) Prove that the curve does not intersect the coordinate axes - AQA - A-Level Maths Mechanics - Question 9 - 2019 - Paper 3

Worked Solution & Example Answer:A curve has equation $$x^2y^2 + xy^4 = 12$$ 9 (a) Prove that the curve does not intersect the coordinate axes - AQA - A-Level Maths Mechanics - Question 9 - 2019 - Paper 3

Prove that the curve does not intersect the coordinate axes.

Show that \(\frac{dy}{dx} = \frac{2xy + y^3}{2x^2 + 4xy^2}\).

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