A-Level AQA Maths Pure Implicit Differentiation: 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 Pure - 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 Pure - Question 9 - 2019 - Paper 3

Step 1

Prove that the curve does not intersect the coordinate axes.

96%

114 rated

Answer

To prove that the curve does not intersect the coordinate axes, we need to show that it does not satisfy the equations for the x-axis and y-axis.

For the x-axis: On the x-axis, we have (y = 0). Substituting (y = 0) into the equation: $x^2(0)^2 + x(0)^4 = 12 \\ 0 = 12$ This results in a contradiction, indicating that the curve does not intersect the x-axis.

For the y-axis: On the y-axis, we have (x = 0). Substituting (x = 0) into the equation: $(0)^2y^2 + 0(y^4) = 12 \\ 0 = 12$ Similarly, this also results in a contradiction. Therefore, the curve does not intersect the y-axis either.

In conclusion, since both intersections yield contradictions, the curve does not intersect the coordinate axes.

Step 2

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

99%

104 rated

Answer

To show the derivative, we use implicit differentiation on the equation $x^2y^2 + xy^4 = 12$ .

Differentiating both sides with respect to (x) gives: $2xy^2 + x(4y^3 \frac{dy}{dx}) + 2y^2 \frac{dy}{dx} = 0.$

Next, we factor out (\frac{dy}{dx}): $x(4y^3)\frac{dy}{dx} + 2y^2\frac{dy}{dx} = -2xy^2.$

This can be rearranged as: $\frac{dy}{dx}(4xy^3 + 2y^2) = -2xy^2.$

Now, solving for (\frac{dy}{dx}): $\frac{dy}{dx} = \frac{-2xy^2}{4xy^3 + 2y^2}.$

This further simplifies to: $\frac{dy}{dx} = \frac{2xy + y^3}{2x^2 + 4xy^2}.$

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 Pure - 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 Pure - Question 9 - 2019 - Paper 3

Prove that the curve does not intersect the coordinate axes.

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

Join the A-Level students using SimpleStudy...