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A ray of light from point P(1, 2) reflects off a point Q on the x-axis and then passes through R(4, 3). Point S(h, k) is such that PQRS forms a parallelogram. Find the value of h*k².

1. 80
2. 90
3. 60
4. 70

Correct answer: 90

Solution

Image of P(1,2) in x-axis: P' = (1,-2). Line P'R: from (1,-2) to (4,3). Slope = (3-(-2))/(4-1) = 5/3. Equation: y+2 = (5/3)(x-1) => y = (5x-5)/3 - 2 = (5x-11)/3. At y=0: 5x=11, x=11/5. So Q = (11/5, 0). For parallelogram PQRS: midpoint of diagonal PR = midpoint of diagonal QS. Midpoint of PR = ((1+4)/2, (2+3)/2) = (5/2, 5/2). Midpoint of QS = ((11/5+h)/2, (0+k)/2) = (5/2, 5/2). So h = 5 - 11/5 = 14/5, k = 5. h*k² = (14/5)*25 = 14*5 = 70.

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