Find the equation of the circle \(x^2+y^2=1\) in terms of \(x'y'\)…
2017
Find the equation of the circle \(x^2+y^2=1\) in terms of \(x'y'\) coordinates, assuming that the \(xy\) coordinate system results from a scaling of 3 units in the \(x'\) direction and 4 units in the \(y'\) direction.
- A.
\(3(x’)^2+4(y’)^2=1\) - B.
\(\bigg( \dfrac{x’}{3} \bigg) ^2 + \bigg( \dfrac{y’}{4} \bigg) ^2 =1\) - C.
\((3x’)^2+ (4y’)^2=1\) - D.
\(\dfrac{1}{3}(x’)^2 + \dfrac{1}{4}(y’)^2=1\)
Attempted by 15 students.
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Correct answer: B
Key idea: express the original coordinates x and y in terms of the scaled coordinates x' and y', then substitute into x^2 + y^2 = 1.
If the xy system results from scaling the x' axis by 3 and the y' axis by 4, then
x = 3x' and y = 4y'.
Substitute into x^2 + y^2 = 1:
(3x')^2 + (4y')^2 = 1
Simplify to get 9(x')^2 + 16(y')^2 = 1.
Therefore, under the standard interpretation that the xy axes are obtained by scaling the x' and y' axes by factors 3 and 4 respectively, the circle's equation in x'y' coordinates is (3x')^2 + (4y')^2 = 1, equivalently 9(x')^2 + 16(y')^2 = 1.
Note: If instead the problem meant that x' = 3x and y' = 4y (the inverse mapping), then the equation becomes (x'/3)^2 + (y'/4)^2 = 1. Clarify the direction of the scaling when describing coordinate transformations.