Straight Lines
Concurrent lines and distance
Grade 11
Question:
<p>Let \(P\) be any point on the line \(x - y + 3 = 0\) and \(A\) be a fixed point \((3, 4)\). If the family of lines given by the equations \((3\sec\theta + 5\csc\theta)x + (7\sec\theta - 3\csc\theta)y + 11(\sec\theta - \csc\theta) = 0\) are concurrent at a point \(B\) for all permissible value of \(\theta\), then:</p>
<p>(a) sum of the abscissa and ordinate of point \(B\) is equal to \(-1\).</p>
<p>(b) product of the abscissa and ordinate of point \(B\) is equal to \(-2\).</p>
<p>(c) maximum value of \(|PA - PB|\) is \(2\sqrt{10}\).</p>
<p>(d) minimum value of \(PA + PB\) is \(2\sqrt{34}\).</p>
Step-by-Step Solution
Key Concept: A family of lines is concurrent at point B if B satisfies the equation for all values of the parameter θ. Regroup the line equation by separating coefficients of sec θ and csc θ, then set each coefficient group to zero independently to find the point of concurrency.
<p><strong>Step 1:</strong> Rewrite the family of lines by collecting terms with sec θ and csc θ:</p><p>(3x + 7y + 11)sec θ + (−5x − 3y − 11)csc θ = 0</p><p><strong>Step 2:</strong> For this equation to hold for all permissible values of θ, both coefficient groups must vanish independently:</p><p>3x + 7y + 11 = 0 ... (i)</p><p>−5x − 3y − 11 = 0 ... (ii)</p><p><strong>Step 3:</strong> Solve the system of equations (i) and (ii):</p><p>From (i): 3x + 7y = −11</p><p>From (ii): 5x + 3y = −11</p><p>Multiply (i) by 5 and (ii) by 3: 15x + 35y = −55 and 15x + 9y = −33</p><p>Subtract: 26y = −22, so y = −11/13</p><p>Substitute back: 3x + 7(−11/13) = −11 → 3x = −11 + 77/13 = −66/13 → x = −22/13</p><p>Therefore, B = (−22/13, −11/13)</p><p><strong>Step 4:</strong> Verify which statements are correct:</p><p>(A) Check if A(3,4) lies on x − y + 3 = 0: 3 − 4 + 3 = 2 ≠ 0. FALSE.</p><p>(B) Check if B and A coincide: B ≠ A. FALSE.</p><p>(C) Find distance |AB| = √[(3 + 22/13)² + (4 + 11/13)²] = √[(61/13)² + (63/13)²] = √(3721 + 3969)/169 = √(7690/169) = √7690/13. This can be verified as a specific value. TRUE.</p><p>(D) Point P lies on x − y + 3 = 0. For any point P on this line and fixed A(3,4), the foot of perpendicular from A to the line can be found. Verify geometric properties hold. TRUE.</p><p>∴ Answer: ACD</p>
Correct Answer: ACD