Describe fully the graph of $x^2 + y^2 = 20.$
The graph of $y = 3x + 10$ intersects the graph of $x^2 + y^2 = 20$ at two points - OCR - GCSE Maths - Question 18 - 2023 - Paper 6
Question 18
Describe fully the graph of $x^2 + y^2 = 20.$
The graph of $y = 3x + 10$ intersects the graph of $x^2 + y^2 = 20$ at two points.
Use an algebraic method to work out... show full transcript
Worked Solution & Example Answer:Describe fully the graph of $x^2 + y^2 = 20.$
The graph of $y = 3x + 10$ intersects the graph of $x^2 + y^2 = 20$ at two points - OCR - GCSE Maths - Question 18 - 2023 - Paper 6
Step 1
Solve $x^2 + y^2 = 20$ for intersection points
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Answer
To find the intersection points of the graphs, substitute the expression for y from the linear equation into the circle equation:
Start with the equation of the circle:
x2+y2=20
Substitute y=3x+10:
x2+(3x+10)2=20
Expand the equation:
x2+(9x2+60x+100)=2010x2+60x+100=2010x2+60x+80=0
Divide the equation by 10:
x2+6x+8=0
Step 2
Factor the quadratic equation
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Answer
Factor or use the quadratic formula:
[x^2 + 6x + 8 = (x + 2)(x + 4) = 0]
Thus, x=−2 or x=−4.
Step 3
Find corresponding y-coordinates
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Answer
Calculate y for both values of x:
For x=−2:
y=3(−2)+10=−6+10=4
Coordinates: (−2,4)
For x=−4:
y=3(−4)+10=−12+10=−2
Coordinates: (−4,−2)
Step 4
Conclusion
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