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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 + ... 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.
Answer
To determine if the curve intersects the coordinate axes, we analyze the equations for when either ( x = 0 ) or ( y = 0 ).
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Substituting ( x = 0 ): This is a contradiction as the left side equals 0, and cannot equal 12.
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Substituting ( y = 0 ): Again, this results in a contradiction, showing that the left side cannot equal 12.
Since both substitutions lead to contradictions, we conclude that the curve does not intersect either axis.
Step 2
Show that \( \frac{dy}{dx} = \frac{2xy + y^3}{2x^2 + 4xy^2} \)
Answer
To find ( \frac{dy}{dx} ) using implicit differentiation, we start from the equation:
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Differentiate both sides implicitly:
- Use the product rule where necessary:
- ( 2xy^2 \frac{dx}{dx} + x(4y^3 \frac{dy}{dx}) + 2x^2y \frac{dy}{dx} + y^4 \frac{dx}{dx} = 0 )
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Substituting ( \frac{dx}{dx} = 1 ):
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Isolate ( \frac{dy}{dx} ):
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Final result: Simplifying the negative signs gives: