Straight Lines
Transformation of Coordinates
Grade 11
Question:
<p>Without changing the direction of coordinate axes, the origin is shifted to <p>(h, k)</p>, then from the equation <p>x^2 + y^2 - 4x + 6y - 7 = 0</p> the terms containing linear powers are missing. Then, point <p>(h, k)</p> is</p>
<p>(a) <p>(3, 2)</p></p>
<p>(b) <p>(-3, 2)</p></p>
<p>(c) <p>(2, -3)</p></p>
<p>(d) <p>(-2, -3)</p></p>
Step-by-Step Solution
Key Concept: Shift origin to <p>(h, k)</p> and require coefficients of linear terms to be zero.
<p><strong>Solution:</strong> When origin is shifted to <p>(h, k)</p>, we substitute <p>x = X + h, y = Y + k</p>.</p><p>The equation becomes: <p>(X + h)^2 + (Y + k)^2 - 4(X + h) + 6(Y + k) - 7 = 0</p></p><p>Expanding: <p>X^2 + Y^2 + 2hX + 2kY + h^2 + k^2 - 4X - 4h + 6Y + 6k - 7 = 0</p></p><p>For linear terms to vanish:</p><p><p>2h - 4 = 0 \implies h = 2</p></p><p><p>2k + 6 = 0 \implies k = -3</p></p><p>Therefore <p>(h, k) = (2, -3)</p></p>
Correct Answer: C