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: Pure - Question 5 - 2019 - Paper 3

Question 5

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... show full transcript

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

Step 1

Find the Radius of the Circle

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Answer

To find the radius of the circle, we start with the standard form of the circle's equation. First, complete the square for the equation:

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

Completing the square:

For $x^2 - 6x$ :
- Take half of -6, square it: $igg(-\frac{6}{2}\bigg)^2 = 9$
- Thus, $x^2 - 6x$ becomes $(x - 3)^2 - 9$ .
For $y^2 - 8y$ :
- Take half of -8, square it: $igg(-\frac{8}{2}\bigg)^2 = 16$
- Thus, $y^2 - 8y$ becomes $(y - 4)^2 - 16$ .

Putting this back into the equation, we have:

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

So,

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

The radius is: $r = \sqrt{289} = 17$

Step 2

Find the Area of the Sector

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Answer

The area of the sector can be found using the formula: $\text{Area of Sector} = \frac{1}{2} r^2 \theta$ where $r = 17$ and $\theta = 0.9$ radians:

$\text{Area of Sector} = \frac{1}{2} \times 17^2 \times 0.9 = \frac{1}{2} \times 289 \times 0.9 = 130.05$

Step 3

Find the Area of the Triangle

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Answer

The area of triangle $OAB$ can be found using: $\text{Area} = \frac{1}{2} r^2 \sin(\theta)$ where $r = 17$ and $\theta = 0.9$ radians:

$\text{Area} = \frac{1}{2} \times 17^2 \times \sin(0.9) = \frac{1}{2} \times 289 \times 0.6216 \approx 89.86$

Step 4

Calculate the Area of the Minor Segment

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Answer

Now, we can find the area of the minor segment by subtracting the area of the triangle from the area of the sector:

$\text{Area of Minor Segment} = \text{Area of Sector} - \text{Area of Triangle}$

Substituting the values: $\text{Area of Minor Segment} = 130.05 - 89.86 \approx 40.19$

Finally, rounding to three significant figures, the area is approximately 40.2.

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

Worked Solution & Example Answer:A circle has equation $x^2 + y^2 - 6x - 8y = 264$ - AQA - A-Level Maths: Pure - 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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