Question:
A ray of light passing through the point P(2, 3) reflects on the x-axis at point A and the reflected ray passes through the point Q(5, 4). Let R be the point that divides the line segment AQ internally into the ratio 2:1. Let the co-ordinates of the foot of the perpendicular M from R on the bisector of the angle PAQ be (α, β). Then, the value of \(7α + 3β\) is equal to ________ .
A ray of light passing through the point P(2, 3) reflects on the x-axis at point A and the reflected ray passes through the point Q(5, 4). Let R be the point that divides the line segment AQ internally into the ratio 2:1. Let the co-ordinates of the foot of the perpendicular M from R on the bisector of the angle PAQ be (α, β). Then, the value of \(7α + 3β\) is equal to ________ .
Updated On: Feb 5, 2026
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Correct Answer: 31
Solution and Explanation
\(\frac {4}{5 - α} = \frac {3}{α - 2}\)
\(⇒ 4α - 8 = 15 - 3α\)
\(α = \frac {23}{7}\)
\(A = ( \frac {23}{7},0)\ Q = (5,4)\)
\(R = (\frac {10 + \frac {23}{7}}{3} , \frac 83 )\)
\(= ( \frac {31}{7} , \frac 83 )\)
Bisector of angle PAQ is \(X = \frac {23}{7}\).
\(⇒ M = ( \frac {23}{7} , \frac 83 )\)
So, \(7α + 3β = 31\)
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Concepts Used:
Tangents and Normals
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m×n = -1
- The normal to a given curve y = f(x) at a point x = x0
- The slope ‘n’ of the normal: As the normal is perpendicular to the tangent, we have: n=-1/m
Diagram Explaining Tangents and Normal:
