Circle C₁ has equation $ \frac{(x-4)^{2}}{37} + \frac{(y+2)^{2}}{37} = 1 $. Circle C₂ has equation $ x^{2} + y^{2} + 2x - 6y - 7 = 0. $ (a) Calculate the distance between the centres of C₁ and C₂. (b) Hence, show that C₁ and C₂ intersect at two distinct points.

Scottish Highers Maths Circle: Circle C₁ has equation \( \frac{

Circle C₁ has equation $ \frac{(x-4)^{2}}{37} + \frac{(y+2)^{2}}{37} = 1 $ - Scottish Highers Maths - Question 11 - 2023

Question 11

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Circle C₁ has equation $ \frac{(x-4)^{2}}{37} + \frac{(y+2)^{2}}{37} = 1 $. Circle C₂ has equation $ x^{2} + y^{2} + 2x - 6y - 7 = 0. $ (a) Calculate the distan... show full transcript

Worked Solution & Example Answer:Circle C₁ has equation $ \frac{(x-4)^{2}}{37} + \frac{(y+2)^{2}}{37} = 1 $ - Scottish Highers Maths - Question 11 - 2023

Step 1

Calculate the distance between the centres of C₁ and C₂.

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Answer

To find the distance between the centres of circles C₁ and C₂, we first need to identify their centres:

Identify the centre of C₁:
- The equation of C₁ can be simplified to identify the centre as ( (4, -2) ).
Identify the centre of C₂:
- We rewrite the equation of C₂ in standard form.
- Completing the square: [ x^{2} + 2x + y^{2} - 6y = 7 ]
  [ (x+1)^{2} - 1 + (y-3)^{2} - 9 = 7 ] [ (x+1)^{2} + (y-3)^{2} = 17 ]
- So, the centre of C₂ is ( (-1, 3) ).
Calculate the distance:
- Now, we apply the distance formula between the centres ( (4, -2) ) and ( (-1, 3) ):
- Using the formula: [ d = \sqrt{(x_2 - x_1)^{2} + (y_2 - y_1)^{2}} ] Where ( (x_1, y_1) = (4, -2) ) and ( (x_2, y_2) = (-1, 3) ).
- This gives us: [ d = \sqrt{((-1) - (4))^{2} + (3 - (-2))^{2}} = \sqrt{(-5)^{2} + (5)^{2}} = \sqrt{25 + 25} = \sqrt{50} \text{ or } 5\sqrt{2}. ]

Step 2

Hence, show that C₁ and C₂ intersect at two distinct points.

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Answer

From part (a), we found that the distance between the centres of the circles is ( d = 5\sqrt{2} \approx 7.07 ).

State the radii of the circles:
- The radius of C₁ can be derived from its equation as ( r_1 = \sqrt{37} \approx 6.08 ).
- The radius of C₂ can be derived from its completed square form as ( r_2 = \sqrt{17} \approx 4.12 ).
Determine the relationship between the distance and the radii:
- We now compare:
  - Distance ( d \approx 7.07 )
  - Radii: ( r_1 + r_2 \approx 6.08 + 4.12 = 10.20 )
  - The distance ( d ) is less than the sum of the radii: [ 7.07 < 10.20 ]
Conclusion:
- Since the distance between the centres is less than the sum of the radii, it shows that the circles C₁ and C₂ intersect at two distinct points.

Circle C₁ has equation \( \frac{(x-4)^{2}}{37} + \frac{(y+2)^{2}}{37} = 1 \) - Scottish Highers Maths - Question 11 - 2023

Worked Solution & Example Answer:Circle C₁ has equation \( \frac{(x-4)^{2}}{37} + \frac{(y+2)^{2}}{37} = 1 \) - Scottish Highers Maths - Question 11 - 2023

Calculate the distance between the centres of C₁ and C₂.

Hence, show that C₁ and C₂ intersect at two distinct points.

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