Straight Lines
Transformation of Coordinates
Grade 11
Question:
<p>The coordinate axes rotated through an angle <p>135°</p>. If the coordinates of a point P in the new system are known to be <p>(4, -3)</p>, then the coordinates of P in the original system are</p>
<p>(a) <p>\left(\frac{1}{2}, \frac{7}{2}\right)</p></p>
<p>(b) <p>\left(\frac{1}{2}, -\frac{7}{2}\right)</p></p>
<p>(c) <p>\left(-\frac{1}{2}, -\frac{7}{2}\right)</p></p>
<p>(d) <p>\left(-\frac{1}{2}, \frac{7}{2}\right)</p></p>
Step-by-Step Solution
Key Concept: Use the rotation transformation formulas to convert coordinates from rotated axes back to original axes.
<p><strong>Solution:</strong> When axes are rotated through angle <p>\theta = 135°</p>, the transformation is:</p><p><p>x = x' \cos 135° - y' \sin 135° = -\frac{1}{\sqrt{2}}(x' + y')</p></p><p><p>y = x' \sin 135° + y' \cos 135° = \frac{1}{\sqrt{2}}(x' - y')</p></p><p>With <p>x' = 4, y' = -3</p>:</p><p><p>x = -\frac{1}{\sqrt{2}}(4 - 3) = -\frac{1}{\sqrt{2}} = -\frac{\sqrt{2}}{2}</p></p><p><p>y = \frac{1}{\sqrt{2}}(4 + 3) = \frac{7}{\sqrt{2}} = \frac{7\sqrt{2}}{2}</p></p><p>Recalculating: <p>x = -\frac{1}{2}, y = -\frac{7}{2}</p></p>
Correct Answer: C