Word Count : 406

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To find the transformation of the given equation under the given rotation of axes while keeping the origin fixed, follow these steps: ### Given equation: \[ 12x^2 - 2y^2 + z^2 = 2xy \] ### Given direction cosines: The new coordinate axes have direction ratios as: - New \( x' \)-axis: \( (1, -3, 0) \) - New \( y' \)-axis: \( (3, 1, 0) \) - New \( z' \)-axis: \( (0, 0, 1) \) #### Step 1: Construct the __________ _____ __________ __________ _______ ________ ________ __________.
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