A-Level Edexcel Maths Pure Inequalities: The curve C₁ with parametric equ

The curve C₁ with parametric equations x = 10cos(t), y = 4√2sin(t), 0 ≤ t < 2π meets the circle C₂ with equation x² + y² = 66 at four distinct points as shown in Figure 2 - Edexcel - A-Level Maths Pure - Question 6 - 2019 - Paper 2

Question 6

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The curve C₁ with parametric equations x = 10cos(t), y = 4√2sin(t), 0 ≤ t < 2π meets the circle C₂ with equation x² + y² = 66 at four distinct points as shown in Fig... show full transcript

Worked Solution & Example Answer:The curve C₁ with parametric equations x = 10cos(t), y = 4√2sin(t), 0 ≤ t < 2π meets the circle C₂ with equation x² + y² = 66 at four distinct points as shown in Figure 2 - Edexcel - A-Level Maths Pure - Question 6 - 2019 - Paper 2

Step 1

Sub-part (i): Establishing the Cartesian Equation

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Answer

To find the Cartesian equation, substitute the parametric equations into the circle's equation:

Substituting:

x = 10cos(t)
y = 4√2sin(t)

The equation becomes:

$(10cos(t))^2 + (4√2sin(t))^2 = 66$

This simplifies to:

$100cos^2(t) + 32sin^2(t) = 66$

Rearranging gives:

$100cos^2(t) + 32sin^2(t) - 66 = 0$

Using the identity $cos^2(t) + sin^2(t) = 1$ , we can express sin² in terms of cos² or vice versa.

Step 2

Sub-part (ii): Solving for Cosine and Sine

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Answer

Rearranging for cosine:

From the previous equation: $32sin^2(t) = 66 - 100cos^2(t)$ Substituting $sin^2(t) = 1 - cos^2(t)$ :

$32(1 - cos^2(t)) = 66 - 100cos^2(t)$ This simplifies to:

$32 - 32cos^2(t) = 66 - 100cos^2(t)$

Bringing all terms involving cos² to one side:

$68cos^2(t) = 34$

Therefore, $cos^2(t) = rac{34}{68} = rac{1}{2}$ Thus, $cos(t) = rac{ rac{1}{2}}{10} = rac{ rac{1}{2}}{10} = rac{1}{20}$

Step 3

Sub-part (iii): Finding the Cartesian Coordinates of S

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Answer

Using $cos(t) = rac{1}{20}$ , find $sin(t)$ using: $sin^2(t) = 1 - cos^2(t) = 1 - rac{1}{400} = rac{399}{400}$ Thus, $sin(t) = - rac{ ext{sqrt}(399)}{20}$ (since S is in the 4th quadrant)

Next, substitute back to find the coordinates of S:

x-coordinate:
$x = 10cos(t) = 10 imes rac{1}{20} = rac{10}{20} = rac{1}{2}$
y-coordinate: $y = 4 ext{\sqrt2}sin(t) = 4 ext{\sqrt2} imes - rac{ ext{sqrt}(399)}{20} = - rac{4 imes \sqrt2 imes \sqrt399}{20}$

Therefore, the coordinates of S are: $S = igg( rac{1}{2}, - rac{4 imes \sqrt2 imes \sqrt399}{20} igg)$

The curve C₁ with parametric equations x = 10cos(t), y = 4√2sin(t), 0 ≤ t < 2π meets the circle C₂ with equation x² + y² = 66 at four distinct points as shown in Figure 2 - Edexcel - A-Level Maths Pure - Question 6 - 2019 - Paper 2

Worked Solution & Example Answer:The curve C₁ with parametric equations x = 10cos(t), y = 4√2sin(t), 0 ≤ t < 2π meets the circle C₂ with equation x² + y² = 66 at four distinct points as shown in Figure 2 - Edexcel - A-Level Maths Pure - Question 6 - 2019 - Paper 2

Sub-part (i): Establishing the Cartesian Equation

Sub-part (ii): Solving for Cosine and Sine

Sub-part (iii): Finding the Cartesian Coordinates of S

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