<p>The point \((x-3, y-3)\) satisfies \(\dfrac{x-3}{\cos(\pi/4)} = \dfrac{y-3}{\sin(\pi/4)} = -2\sqrt{2}\). Find the new position of the point.</p>
Step-by-Step Solution
Key Concept: Recognize that the equation represents a parametric form where a point is displaced from (3,3) along a direction making angle π/4 with x-axis by distance 2√2 in the opposite direction. Use the parametric relationships to find actual (x,y) coordinates.
<p><strong>Step 1:</strong> Recognize the parametric form. Let (x-3) = X and (y-3) = Y. Then:</p><p>$$\frac{X}{\cos(\pi/4)} = \frac{Y}{\sin(\pi/4)} = -2\sqrt{2}$$</p><p><strong>Step 2:</strong> Apply the parametric relationships:</p><p>$$X = -2\sqrt{2} \cdot \cos(\pi/4) = -2\sqrt{2} \cdot \frac{1}{\sqrt{2}} = -2$$</p><p>$$Y = -2\sqrt{2} \cdot \sin(\pi/4) = -2\sqrt{2} \cdot \frac{1}{\sqrt{2}} = -2$$</p><p><strong>Step 3:</strong> Since X = x-3 and Y = y-3, we have:</p><p>$$x - 3 = -2 \Rightarrow x = 1$$</p><p>$$y - 3 = -2 \Rightarrow y = 1$$</p><p><strong>Step 4:</strong> The new position of the point is (1, 1).</p><p>∴ Answer: A (Assuming option A is (1, 1))</p>
Correct Answer: A