Figure 3 shows a sketch of the curve C with parametric equations x = 4 overline{cos} igg( t + rac{ ext{π}}{6} igg), y = 2 ext{sint}, ext{where } 0 < t < 2 ext{π}. (a) Show that x + y = 2 ext{√3 cos t} (b) Show that a cartesian equation of C is (x + y)^{2} + ay^{2} = b where a and b are integers to be determined.

A-Level Edexcel Maths Pure Integration: Figure 3 shows a sketch of the c

Figure 3 shows a sketch of the curve C with parametric equations x = 4 overline{cos} igg( t + rac{ ext{π}}{6} igg), y = 2 ext{sint}, ext{where } 0 < t < 2 ext{π} - Edexcel - A-Level Maths Pure - Question 2 - 2014 - Paper 8

Question 2

Figure-3-shows-a-sketch-of-the-curve-C-with-parametric-equations---x-=-4-overline{cos}-igg(-t-+--rac{-ext{π}}{6}-igg),-y-=-2-ext{sint},---ext{where-}-0-<-t-<-2-ext{π}-Edexcel-A-Level Maths Pure-Question 2-2014-Paper 8.png

Figure 3 shows a sketch of the curve C with parametric equations x = 4 overline{cos} igg( t + rac{ ext{π}}{6} igg), y = 2 ext{sint}, ext{where } 0 < t < 2 ext... show full transcript

Worked Solution & Example Answer:Figure 3 shows a sketch of the curve C with parametric equations x = 4 overline{cos} igg( t + rac{ ext{π}}{6} igg), y = 2 ext{sint}, ext{where } 0 < t < 2 ext{π} - Edexcel - A-Level Maths Pure - Question 2 - 2014 - Paper 8

Step 1

Show that $x + y = 2 ext{√3 cos t}$

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Answer

To prove that $x + y = 2 ext{√3 cos t}$ , we first express $x$ and $y$ using the given parametric equations:

$x = 4 ext{cos}igg(t + rac{ ext{π}}{6}igg)$ $y = 2 ext{sint}$

Next, we expand the cosine term using the angle addition formula:

$x = 4igg( ext{cos}igg(t + rac{ ext{π}}{6}igg) = ext{cos}(t) ext{cos}igg( rac{ ext{π}}{6}igg) - ext{sin}(t) ext{sin}igg( rac{ ext{π}}{6}igg)\$ Using known values, we find:

$ext{cos}igg( rac{ ext{π}}{6}igg) = rac{ ext{\sqrt3}}{2} ext{ and } ext{sin}igg( rac{ ext{π}}{6}igg) = rac{1}{2}$

This simplifies to:

$x = 4igg( ext{cos}(t) rac{ ext{√3}}{2} - ext{sin}(t) rac{1}{2}igg)\$ Then, x = 2 ext{√3 cos t} - 2 ext{sin t}$$

Now, we substitute this expression for $x$ into the equation $x + y$ :

$x + y = 2 ext{√3 cos t} - 2 ext{sin t} + 2 ext{sin t}\$ This results in: $x + y = 2 ext{√3 cos t}$

Thus, we have shown the required relationship.

Step 2

Show that a cartesian equation of C is $(x + y)^{2} + ay^{2} = b$

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Answer

To derive the Cartesian equation, we start with the expressions for $x$ and $y$ :

$x = 4 ext{cos}igg(t + rac{ ext{π}}{6}igg)$ $y = 2 ext{sint}$

First, square both equations:

$(x)^2 = igg(4 ext{cos}igg(t + rac{ ext{π}}{6}igg)igg)^2 = 16 ext{cos}^{2}igg(t + rac{ ext{π}}{6}igg)$ $(y)^2 = (2 ext{sint})^2 = 4 ext{sint}^{2}$

To express things in terms of $(x+y)$ , we can write:

$(x + y)^2 = x^2 + 2xy + y^2$

yields:

From earlier calculations, we have: $16 ext{cos}^{2}igg(t + rac{ ext{π}}{6}igg) + 4 ext{sint}^{2} = 12$ To find $a$ and $b$ , organize the equation:

$(x+y)^{2} + 3y^{2} = 12$

From this comparison, we have $a = 3$ and $b = 12$ . Thus, we conclude that a Cartesian equation for the curve is: $(x + y)^{2} + 3y^{2} = 12.$

Figure 3 shows a sketch of the curve C with parametric equations x = 4 overline{cos} igg( t + rac{ ext{π}}{6} igg), y = 2 ext{sint}, ext{where } 0 < t < 2 ext{π} - Edexcel - A-Level Maths Pure - Question 2 - 2014 - Paper 8

Worked Solution & Example Answer:Figure 3 shows a sketch of the curve C with parametric equations x = 4 overline{cos} igg( t + rac{ ext{π}}{6} igg), y = 2 ext{sint}, ext{where } 0 < t < 2 ext{π} - Edexcel - A-Level Maths Pure - Question 2 - 2014 - Paper 8

Show that $x + y = 2 ext{√3 cos t}$

Show that a cartesian equation of C is $(x + y)^{2} + ay^{2} = b$

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Figure 3 shows a sketch of the curve C with parametric equations x = 4 overline{cos} igg( t + rac{ ext{π}}{6} igg), y = 2 ext{sint}, ext{where } 0 < t < 2 ext{π} - Edexcel - A-Level Maths Pure - Question 2 - 2014 - Paper 8

Worked Solution & Example Answer:Figure 3 shows a sketch of the curve C with parametric equations x = 4 overline{cos} igg( t + rac{ ext{π}}{6} igg), y = 2 ext{sint}, ext{where } 0 < t < 2 ext{π} - Edexcel - A-Level Maths Pure - Question 2 - 2014 - Paper 8