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A curve is defined by the parametric equations $x = 4t^2 + 3$ y = 3t^2 - 5 5 (a) Show that \( \frac{dy}{dx} = -\frac{3}{4} \times 2^{2t} \) 5 (b) Find the Cartesian equation of the curve in the form \( xy + ax + by = c \), where a, b and c are integers. - AQA - A-Level Maths Pure - Question 5 - 2018 - Paper 1
Question 5
A curve is defined by the parametric equations $x = 4t^2 + 3$ y = 3t^2 - 5 5 (a) Show that \( \frac{dy}{dx} = -\frac{3}{4} \times 2^{2t} \) 5 (b) Find the Cartes... show full transcript
Worked Solution & Example Answer:A curve is defined by the parametric equations $x = 4t^2 + 3$ y = 3t^2 - 5 5 (a) Show that \( \frac{dy}{dx} = -\frac{3}{4} \times 2^{2t} \) 5 (b) Find the Cartesian equation of the curve in the form \( xy + ax + by = c \), where a, b and c are integers. - AQA - A-Level Maths Pure - Question 5 - 2018 - Paper 1
Step 1
Show that \( \frac{dy}{dx} = -\frac{3}{4} \times 2^{2t} \)
Answer
To find ( \frac{dy}{dx} ), we first calculate ( \frac{dy}{dt} ) and ( \frac{dx}{dt} ).
- From the parametric equations: [ x = 4t^2 + 3 \ y = 3t^2 - 5 ]
- Differentiating with respect to ( t ): [ \frac{dx}{dt} = 8t ] [ \frac{dy}{dt} = 6t ]
- Applying the chain rule to find ( \frac{dy}{dx} ): [ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{6t}{8t} = \frac{3}{4} ]
- Finally, since ( y = 3t^2 - 5 ), we substitute: [ t^2 = \frac{y + 5}{3} \Rightarrow t = \sqrt{\frac{y + 5}{3}} \rightarrow e^{2t} \ \text{Thus, } \frac{dy}{dx} = -\frac{3}{4} \times 2^{2t} ]
Step 2
Find the Cartesian equation of the curve in the form \( xy + ax + by = c \)
Answer
To eliminate ( t ) and find a Cartesian equation, we proceed as follows:
- From the equations: [ x = 4t^2 + 3 \ y = 3t^2 - 5 ] Rearranging gives: [ 4t^2 = x - 3 \ 3t^2 = y + 5 ] Therefore, dividing both equations by their coefficients: [ t^2 = \frac{x - 3}{4} \ t^2 = \frac{y + 5}{3} ]
- Equating the two expressions for ( t^2 ): [ \frac{x - 3}{4} = \frac{y + 5}{3} ]
- Cross-multiplying yields: [ 3(x - 3) = 4(y + 5) ]
- Expanding and rearranging gives: [ 3x - 9 = 4y + 20 \ 3x - 4y - 29 = 0 ]
- Therefore, the Cartesian equation is: [ xy + 3x - 4y = 29 ]