A circle has equation $x^{2} + y^{2} - 6x - 8y = 264$. $AB$ is a chord of the circle. The angle at the centre of the circle, subtended by $AB$, is 0.9 radians, as shown in the diagram below. Find the area of the minor segment shaded on the diagram. Give your answer to three significant figures.

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A circle has equation $x^{2} + y^{2} - 6x - 8y = 264$ - AQA - A-Level Maths Mechanics - Question 5 - 2019 - Paper 3

Question 5

$A-circle-has-equation-$x^{2}-+-y^{2}---6x---8y-=-264$-AQA-A-Level Maths Mechanics-Question 5-2019-Paper 3.png$

A circle has equation $x^{2} + y^{2} - 6x - 8y = 264$. $AB$ is a chord of the circle. The angle at the centre of the circle, subtended by $AB$, is 0.9 radians, as ... show full transcript

Worked Solution & Example Answer:A circle has equation $x^{2} + y^{2} - 6x - 8y = 264$ - AQA - A-Level Maths Mechanics - Question 5 - 2019 - Paper 3

Step 1

Find the radius of the circle

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Answer

To find the radius, we first rewrite the circle equation in the standard form by completing the square. The given equation is:

$x^{2} + y^{2} - 6x - 8y = 264$

Completing the square:

For $x^{2} - 6x$ :

$x^{2} - 6x = (x - 3)^{2} - 9$
For $y^{2} - 8y$ :

$y^{2} - 8y = (y - 4)^{2} - 16$

Now substituting back:

$(x - 3)^{2} - 9 + (y - 4)^{2} - 16 = 264$

This simplifies to:

$(x - 3)^{2} + (y - 4)^{2} = 289$

Thus, the radius is:

$r = ext{sqrt}(289) = 17$

Step 2

Find the area of the sector

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Answer

The area of the sector can be calculated using the formula:

$ext{Area of Sector} = rac{1}{2} r^{2} heta$

Substituting the values we found:

$r = 17$
$heta = 0.9$

So,

$ext{Area of Sector} = rac{1}{2} imes 17^{2} imes 0.9 = rac{1}{2} imes 289 imes 0.9 = 130.05$

Step 3

Find the area of the triangle

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Answer

The area of triangle $OAB$ can be found using:

$ext{Area of Triangle} = rac{1}{2} r^{2} ext{sine}( heta)$

Using $r = 17$ and $heta = 0.9$ :

$ext{Area of Triangle} = rac{1}{2} imes 17^{2} imes ext{sin}(0.9)$

Calculating this we use:

$ext{sin}(0.9) ext{ (approximately } 0.7833 ext{)}$

So,

$ext{Area of Triangle} = rac{1}{2} imes 289 imes 0.7833 ext{ (approximately equal to } 113.19)$

Step 4

Calculate the area of the minor segment

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Answer

The area of the minor segment is given by:

$ext{Area of Minor Segment} = ext{Area of Sector} - ext{Area of Triangle}$

Now substituting the previously calculated areas:

$ext{Area of Minor Segment} = 130.05 - 113.19 = 16.86$

Thus, rounded to three significant figures, the final answer is:

$ext{Area of Minor Segment} = 16.9$

A circle has equation $x^{2} + y^{2} - 6x - 8y = 264$ - AQA - A-Level Maths Mechanics - Question 5 - 2019 - Paper 3

Worked Solution & Example Answer:A circle has equation $x^{2} + y^{2} - 6x - 8y = 264$ - AQA - A-Level Maths Mechanics - Question 5 - 2019 - Paper 3

Find the radius of the circle

Find the area of the sector

Find the area of the triangle

Calculate the area of the minor segment

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