<p>Suppose <span>A(1, 1)</span> and <span>B(2, -3)</span> are two points and <span>D</span> is a point on <span>AB</span> produced such that <span>AD = 3AB</span>. Then, coordinates of <span>D</span> are</p>
Step-by-Step Solution
Key Concept: Recognize that when D is on AB produced with AD = 3AB, point D divides AB externally in the ratio 3:2. Apply the external division formula to find the coordinates.
<p><strong>Given:</strong> <span>AD = 3AB</span></p><p><strong>Step 1:</strong> Since <span>D</span> is on <span>AB</span> produced with <span>AD = 3AB</span>, we have <span>BD = AD - AB = 3AB - AB = 2AB</span></p><p><strong>Step 2:</strong> Thus, <span>D</span> divides <span>AB</span> externally in the ratio <span>AD : BD = 3 : 2</span></p><p><strong>Step 3:</strong> Using the external division formula, if point <span>P(x,y)</span> divides the line segment joining <span>A(x_1, y_1)</span> and <span>B(x_2, y_2)</span> in the ratio <span>m:n</span> externally, then:</p><p>$$x = \frac{mx_2 - nx_1}{m - n}, \quad y = \frac{my_2 - ny_1}{m - n}$$</p><p><strong>Step 4:</strong> Substituting <span>A(1,1)</span>, <span>B(2,-3)</span>, <span>m = 3</span>, and <span>n = 2</span>:</p><p>$$x = \frac{3(2) - 2(1)}{3 - 2} = \frac{6 - 2}{1} = 4$$</p><p>$$y = \frac{3(-3) - 2(1)}{3 - 2} = \frac{-9 - 2}{1} = -11$$</p><p><strong>∴ The coordinates of D are (4, -11)</strong></p>
Correct Answer: b