Q5. Q moves on the line y = 2x + 1. Find the locus of the mid-point of OQ, where O is the origin.
If the mid-point is (h, k), then Q = (2h, 2k). Q is on the line, so 2k = 4h + 1. The locus is 4x − 2y + 1 = 0, that is y = 2x + 1/2.
Q6. Find the locus of P such that P, A(1, 2) and B(3, 6) are collinear.
The area of triangle PAB must be zero: h(2 − 6) + 1(6 − k) + 3(k − 2) = −4h + 2k = 0. The locus is y = 2x, the line AB itself.
Q7. A triangle has vertices O(0, 0), A(3, 0) and B, where B moves on the line y = 6. Find the locus of its centroid, the point ((x1 + x2 + x3)/3, (y1 + y2 + y3)/3).
With B = (t, 6), the centroid is ((3 + t)/3, 6/3) = ((3 + t)/3, 2). Its y-coordinate is always 2, and every x is possible, so the locus is the line y = 2.
Q8. A rod AB of length 10 slides with A on the x-axis and B on the y-axis. Find the locus of the point P of AB with AP : PB = 2 : 3.
Let A = (a, 0) and B = (0, b), with a2 + b2 = 100. By the section formula, P = ((2 × 0 + 3a)/5, (2b + 3 × 0)/5) = (3a/5, 2b/5). So a = 5h/3 and b = 5k/2, and 25h2/9 + 25k2/4 = 100.
The locus is x2/36 + y2/16 = 1, a curve called an ellipse. NCERT’s Conic Sections chapter works a sliding-rod locus of the same kind with a 15 cm rod (Chapter 10, Miscellaneous Example 19, page 203).
Q9. If the sum of the perpendicular distances of a variable point P(x, y) from the lines x + y − 5 = 0 and 3x − 2y + 7 = 0 is always 10, show that P must move on a line (NCERT Miscellaneous Exercise, Q19).
The condition is |x + y − 5|/√2 + |3x − 2y + 7|/√13 = 10. If P stays on one side of each line, each bracket keeps its sign. For example, with both positive:
(x + y − 5)/√2 + (3x − 2y + 7)/√13 = 10.
This equation is of the first degree in x and y, so P moves on a line.
So NCERT’s statement holds only while P stays on one side of each line. Over the whole plane, the four choices of sign give four such lines. The points that satisfy the condition form the four sides of a rectangle centred at (3/5, 22/5), the point where the two given lines cross; its diagonals lie along the two given lines.
Q10. Find the locus of a point P that moves so that its distance from (1, 0) is twice its distance from the y-axis.
(h − 1)2 + k2 = 4h2. The locus is 3x2 − y2 + 2x − 1 = 0, a curve called a hyperbola. For example, (1/3, 0) is on it: it is 2/3 from (1, 0) and 1/3 from the y-axis.