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Questions and Answers
What is the general form of the equation of a circle in this context?
Which point is used to find the position of the center C of the circle?
What does the equation of the line representing the center of the circle look like?
After substituting point B(6,5) into the circle equation, what expression is obtained?
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What value do you solve for when substituting in point A(4,1) into the equation of the circle?
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Study Notes
Finding the Equation of a Circle
- Circle equation format: (x^2+y^2+2gx+2fy+c=0).
- Centre of the circle is at (C(g, f)).
Conditions Given
- Circle passes through points (A(4,1)) and (B(6,5)).
- Centre lies on the line given by the equation (4x+3y-24=0).
Condition from Line Equation
- Substituting point (C(g,f)) into the line equation provides:
- (4g + 3f - 24 = 0).
Substituting Points into Circle Equation
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For point (A(4,1)):
- Substituting values into the circle equation yields:
- (16 + 1 + 8g + 2f + c = 0 \rightarrow 8g + 2f + c + 17 = 0).
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For point (B(6,5)):
- Substituting values also into the circle equation gives:
- (36 + 25 + 12g + 10f + c = 0 \rightarrow 12g + 10f + c + 61 = 0).
Summary of Equations
- From line condition: (4g + 3f - 24 = 0).
- Circle equation conditions:
- (8g + 2f + c + 17 = 0).
- (12g + 10f + c + 61 = 0).
Next Steps
- Solve the resulting system of equations to find values of (g), (f), and (c).
- Combine results to formulate the final equation of the circle.
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Description
This quiz focuses on finding the equation of a circle that passes through specific points and has its center on a given line. Students will apply concepts from coordinate geometry and algebra to deduce the correct circle equation based on the conditions provided. Perfect for students looking to enhance their understanding of circle equations.
