Find the exact coordinates of the two intersections of the line $y = 2x$ and the circle $x^2 + y^2 = 30$. - OCR - GCSE Maths - Question 17 - 2018 - Paper 6
Question 17
Find the exact coordinates of the two intersections of the line $y = 2x$ and the circle $x^2 + y^2 = 30$.
Worked Solution & Example Answer:Find the exact coordinates of the two intersections of the line $y = 2x$ and the circle $x^2 + y^2 = 30$. - OCR - GCSE Maths - Question 17 - 2018 - Paper 6
Step 1
Substitute the line equation into the circle equation
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Answer
To find the points of intersection, substitute y=2x into the circle's equation:
x2+(2x)2=30
This simplifies to:
x2+4x2=30
or
5x2=30
Step 2
Solve for x
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Answer
Now solve for x:
x2=6
Taking the square root gives us two possible values:
x = rac{ ext{sqrt}(6)}{1} ext{ and } x = - rac{ ext{sqrt}(6)}{1}
Step 3
Find corresponding y coordinates
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Answer
Next, substitute these x values back into the line equation to find the corresponding y values:
For x=extsqrt(6):
y=2(extsqrt(6))=2extsqrt(6)
For x=−extsqrt(6):
y=2(−extsqrt(6))=−2extsqrt(6)
Step 4
Provide the final coordinate answers
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Answer
Thus, the exact coordinates of the two intersections are:
