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Difficulty: HardLoci and Geometric Constructions

A point P(x,y)P(x, y) moves such that it is equidistant from two parallel lines 3x−4y+11=03x - 4y + 11 = 0 and 3x−4y−1=03x - 4y - 1 = 0, defining locus L1L_1. A second locus L2L_2 consists of all points that are at a constant distance of 55 units from the fixed point (1,7)(1, 7). Calculate the distance between the two points of intersection of locus L1L_1 and locus L2L_2.

Answer: 6 units

Answer

The distance between the two points of intersection of locus L1 and locus L2 is 6 units.
Locus L1 is the line midway between the two given parallel lines, having the equation 3x - 4y + 5 = 0. Locus L2 is a circle centered at (1, 7) with radius 5. The perpendicular distance from the center (1, 7) to line L1 is 4 units. Applying Pythagoras' theorem to the right triangle formed by the radius, perpendicular distance, and half-chord gives a half-chord length of sqrt(5^2 - 4^2) = 3. Therefore, the distance between the two intersection points (the full chord length) is 2 * 3 = 6 units.

Step-by-Step Solution

1
Determine the equation of locus L1
3x - 4y + 5 = 0
The locus of points equidistant from two parallel lines ax + by + c1 = 0 and ax + by + c2 = 0 is a line parallel to both and midway between them, given by ax + by + (c1 + c2)/2 = 0.
2
Identify the geometric representation of locus L2
A circle centered at (1, 7) with radius r = 5
The locus of points at a constant distance r from a fixed point (x0, y0) is a circle.
3
Calculate the perpendicular distance d from the circle's center (1, 7) to line L1
d = 4 units
Using the perpendicular distance formula d = |ax0 + by0 + c| / sqrt(a^2 + b^2), we get d = |3(1) - 4(7) + 5| / sqrt(3^2 + (-4)^2) = |-20| / 5 = 4.
4
Compute the total distance between the two intersection points
6 units
The line intersects the circle along a chord. Using Pythagoras' theorem, half the chord length is sqrt(r^2 - d^2) = sqrt(5^2 - 4^2) = 3. Multiplying by 2 gives the full distance of 6 units.

Key Concept

Intersection of Loci (Parallel Line Bisector and Circle)
Estimated Time:2m 30s
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Topics

Loci and Geometric Constructions

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