Question:
If p and q are real numbers between 0 and 1 such that the points (p, 1), (1, q) and (0, 0) form an equilateral triangle, then what is (p + q) equal to?
If p and q are real numbers between 0 and 1 such that the points (p, 1), (1, q) and (0, 0) form an equilateral triangle, then what is (p + q) equal to?
Options:
- a) 2
- b) 21-
- c) 23-
- d) 42 3-
- a) 2
- b) 21-
- c) 23-
- d) 42 3-
correct answer : d)
Explanation:
p2 1+ = q2 1+ = () ()pq-+ -1122
p2 1+ = q2 1+ Þ p = q [\ p, q Î (0, 1)]
Now, q2 1+ = pq-() +-112 2()
Þ p2 + 1 = 2 (p – 1)2
Þ p2 – 4p + 1 = 0 Þ p = 23±
Since, 0 < p < 1 Þ p = 23- = q
Now, p + q = 22 3-() = 42 3-
p2 1+ = q2 1+ = () ()pq-+ -1122
p2 1+ = q2 1+ Þ p = q [\ p, q Î (0, 1)]
Now, q2 1+ = pq-() +-112 2()
Þ p2 + 1 = 2 (p – 1)2
Þ p2 – 4p + 1 = 0 Þ p = 23±
Since, 0 < p < 1 Þ p = 23- = q
Now, p + q = 22 3-() = 42 3-