Hint
Differentiate y as a function of x. Collect \(\displaystyle (ax+2y)y'=-(2x+ay)\).
Worked solution
(a)
Differentiate to get 2x+a(y+xy′)+2yy′=0, then collect and divide where \(\displaystyle ax+2y\ne 0\).
Answer: \(\displaystyle \frac{dy}{dx}=-\frac{2x+3y}{3x+2y}\)
(b)
The point satisfies the implicit equation. Its derivative is \(\displaystyle - \frac{7}{8}\), and the tangent passes through \(\left( 4, \ 2\right)\).
Answer: \(\displaystyle y=\frac{11}{2} - \frac{7 x}{8}\)
(c)
Take the negative reciprocal of the finite nonzero tangent gradient.
\(\displaystyle \frac{8}{7}\)
Answer: \(\displaystyle \frac{8}{7}\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit complete valid alternatives and consequential reasoning; an unexplained final answer does not earn a method mark.
(a) · 2 marks
- 1 mark: Differentiate each term, including both product-rule terms, and isolate y′ where its denominator is nonzero.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: The xy term needs the product rule.
(b) · 2 marks
- 1 mark: Evaluate the derivative at the stated point and form its tangent equation.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: A zero denominator may indicate a vertical tangent.
(c) · 2 marks
- 1 mark: Take the negative reciprocal of the finite nonzero tangent gradient.
- 1 mark: Give the correct requested result, retaining signs, units and any domain restrictions.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: The xy term needs the product rule.
Total: 6 marks.