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Figure 2 shows a sketch of the curve C with parametric equations $x = 4 an t$, $y = 5 rac{ oot{3}}{2} ext{sin} 2t$, $0 leq t leq rac{ ext{π}}{2}$ - Edexcel - A-Level Maths Pure - Question 7 - 2016 - Paper 4
Question 7
Figure 2 shows a sketch of the curve C with parametric equations $x = 4 an t$, $y = 5 rac{ oot{3}}{2} ext{sin} 2t$, $0 leq t leq rac{ ext{π}}{2}$. The po... show full transcript
Worked Solution & Example Answer:Figure 2 shows a sketch of the curve C with parametric equations $x = 4 an t$, $y = 5 rac{ oot{3}}{2} ext{sin} 2t$, $0 leq t leq rac{ ext{π}}{2}$ - Edexcel - A-Level Maths Pure - Question 7 - 2016 - Paper 4
Step 1
Find the exact value of $ rac{dy}{dx}$ at the point P.
Answer
To find rac{dy}{dx}, we will use the parametric equations given. First, we find rac{dy}{dt} and rac{dx}{dt}:
- Differentiate the equations with respect to :
- For , we have:
- For y = 5 rac{ oot{3}}{2} ext{sin} 2t, we use the chain rule:
oot{3}}{2} imes 2 ext{cos} 2t = 5 oot{3} ext{cos} 2t$$
- Then, we can find rac{dy}{dx} using the formula:
oot{3} ext{cos} 2t}{4 ext{sec}^2 t} = rac{5 oot{3}}{4} rac{ ext{cos} 2t}{ ext{cos}^2 t}$$
- Next, we need to evaluate this at the given point P. Calculate for point P:
- Given the coordinates of point P, we can find by solving x = 4 an t = 4 rac{ oot{3}}{3}, leading to:
oot{3}}{3} ightarrow t = rac{ ext{π}}{6}$$
- Substitute back into the expression for rac{dy}{dx}:
oot{3}}{4} rac{ ext{cos}( rac{ ext{π}}{3})}{ ext{cos}^2( rac{ ext{π}}{6})} = rac{5 oot{3}}{4} rac{ rac{1}{2}}{ rac{3}{4}} = rac{5 oot{3}}{4} imes rac{2}{3} = rac{10 oot{3}}{12} = rac{5 oot{3}}{6}$$
Step 2
Find the exact coordinates of the point Q.
Answer
To find the coordinates of the point Q where rac{dy}{dx} = 0, we first recall that:
oot{3}}{2} imes 2 ext{cos} 2t$$ Setting $ rac{dy}{dt} = 0$ gives: $$ ext{cos} 2t = 0$$ This happens when: $$2t = rac{ ext{π}}{2} + n ext{π}, ext{ for } n ext{ integer}$$ This gives: $$t = rac{ ext{π}}{4}, rac{3 ext{π}}{4}$$ For $t = rac{ ext{π}}{4}$, substitute $t$ into the equations for $x$ and $y$: 1. For $x$: $$x = 4 an rac{ ext{π}}{4} = 4 imes 1 = 4$$ 2. For $y$: $$y = 5 rac{ oot{3}}{2} ext{sin} 2( rac{ ext{π}}{4}) = 5 rac{ oot{3}}{2} imes 1 = rac{5 oot{3}}{2}$$ Thus, the coordinates of point Q are: $$(4, rac{5 oot{3}}{2})$$