<p>Given A (1, 1) and AB is any line through it cutting the x-axis in B. If AC is perpendicular to AB and meets the y-axis in C, then the equation of locus of mid-point P of BC is:</p>
Step-by-Step Solution
Key Concept: Use parametric representation of line AB through A(1,1), find where it intersects axes, then use the perpendicularity condition to locate C. Finally, find the locus of the midpoint P of BC.
<p><strong>Step 1: Parametrize line AB</strong></p><p>Let AB have slope m (where m ≠ 0 since AB cuts the x-axis and passes through A(1,1)).</p><p>Equation of AB: y - 1 = m(x - 1)</p><p><strong>Step 2: Find point B (intersection with x-axis)</strong></p><p>Set y = 0: -1 = m(x - 1) → x = 1 - 1/m</p><p>So B = (1 - 1/m, 0)</p><p><strong>Step 3: Find line AC (perpendicular to AB)</strong></p><p>Since AC ⊥ AB, slope of AC = -1/m</p><p>Equation of AC: y - 1 = (-1/m)(x - 1)</p><p><strong>Step 4: Find point C (intersection with y-axis)</strong></p><p>Set x = 0: y - 1 = (-1/m)(0 - 1) = 1/m</p><p>So y = 1 + 1/m</p><p>Thus C = (0, 1 + 1/m)</p><p><strong>Step 5: Find midpoint P of BC</strong></p><p>Let P = (h, k) be the midpoint of B and C:</p><p>h = [(1 - 1/m) + 0]/2 = (1 - 1/m)/2 = 1/2 - 1/(2m)</p><p>k = [0 + (1 + 1/m)]/2 = (1 + 1/m)/2 = 1/2 + 1/(2m)</p><p><strong>Step 6: Eliminate the parameter m</strong></p><p>From h: 1/(2m) = 1/2 - h</p><p>From k: 1/(2m) = k - 1/2</p><p>Therefore: 1/2 - h = k - 1/2</p><p>1/2 - h = k - 1/2</p><p>1 = h + k</p><p>h + k = 2</p><p><strong>Step 7: Replace (h,k) with (x,y)</strong></p><p>The locus equation is: x + y = 2</p><p><strong>∴ Answer:</strong> B</p>
Correct Answer: B