A-Level AQA Maths Mechanics Quantities, Units & Modelling: 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 $rac{dy}{dx} = \frac{2xy + y^3}{... 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 must prove that there are no points on the curve where either $x = 0$ or $y = 0$ .

Substituting $x = 0$ :
- Substitute into the equation: $0^2y^2 + 0 imes y^4 = 12$ .
- This simplifies to $0 = 12$ , which is a contradiction.
Substituting $y = 0$ :
- Substitute into the equation: $x^2(0^2) + x(0^4) = 12$ .
- This simplifies to $0 = 12$ , which is also a contradiction.

Since both cases lead to contradictions, we conclude that the curve does not intersect the coordinate axes.

Step 2

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

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Answer

Implicit Differentiation:
- Start with the equation: $x^2y^2 + xy^4 = 12$ .
- Differentiate both sides with respect to $x$ $x$ using the product rule and implicit differentiation:
  - Differentiate $x^2y^2$ : $2xy^2 + x^2 \frac{d(y^2)}{dx}$ .
  - Recall: $\frac{d(y^2)}{dx} = 2y \frac{dy}{dx}$ .
  - Thus: $2xy^2 + 2x^2y \frac{dy}{dx}$ .
- Differentiate $xy^4$ : $y^4 + 4xy^3 \frac{dy}{dx}$ .
- Differentiate the right-hand side: $0$ .
Combine and Collect Terms:
- Now, we have:
  
  $2xy^2 + 2x^2y \frac{dy}{dx} + y^4 + 4xy^3 \frac{dy}{dx} = 0$
- Rearranging terms yields:
  
  $\left(2x^2y + 4xy^3\right) \frac{dy}{dx} = - (2xy^2 + y^4)$
Solve for ( \frac{dy}{dx} ):
- Dividing both sides by $(2x^2 + 4xy^2)$ 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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