Question

If the coordinate axes are rotated by \( 45^\circ \) about the origin in the counterclockwise direction, then the transformed equation of \( y^2 = 4ax \) is:

• A. \( (x+y)^2 = 4\sqrt{2}a(x - y) \)
• B. \( (x-y)^2 = 4\sqrt{2}a(x+y) \)
• C. \( (x-y)^2 = \frac{4a}{\sqrt{2}}(x+y) \)
• D. \( (x+y)^2 = \frac{4a}{\sqrt{2}}(x-y) \)

Hint:

Rotation transformations use trigonometric functions to shift coordinate systems.

Answer 1

The Correct Option is A

Step 1: Rotation Transformation Equations
\[ x = X \cos 45^\circ - Y \sin 45^\circ \] \[ y = X \sin 45^\circ + Y \cos 45^\circ \] Substituting \( \cos 45^\circ = \sin 45^\circ = \frac{1}{\sqrt{2}} \): \[ x = \frac{X - Y}{\sqrt{2}}, \quad y = \frac{X + Y}{\sqrt{2}} \] Step 2: Transforming \( y^2 = 4ax \)
\[ \left( \frac{X+Y}{\sqrt{2}} \right)^2 = 4a \left( \frac{X - Y}{\sqrt{2}} \right) \] Multiplying both sides by 2: \[ (X+Y)^2 = 4\sqrt{2} a (X - Y) \] Thus, the correct answer is \( (x+y)^2 = 4\sqrt{2}a(x - y) \).