Answer:
To solve this problem, we can use the Pythagorean Theorem to solve for a.
For part (a), we know that P(2, 3) and Q(a, -1) are 4 units apart, so we can use the Pythagorean Theorem to calculate a:
(a - 2)² + (-1 - 3)² = 4²
Solving for a, we get:
a = 7
For part (b), we know that P(-1, 1) and Q(a, -2) are 5 units apart, so we can use the Pythagorean Theorem to calculate a:
(a + 1)² + (-2 - 1)² = 5²
Solving for a, we get:
a = 4
For part (c), we know that X(a, a) is √√8 units from the origin, so we can use the Pythagorean Theorem to calculate a:
a² + a² = 8
Solving for a, we get:
a = 2√2
For part (d), we know that A(0, a) is equidistant from P(3, -3) and Q(-2, 2), so we can use the Pythagorean Theorem to calculate a:
(3 - 0)² + (-3 - a)² = (-2 - 0)² + (2 - a)²
Solving for a, we get:
a = -1
Therefore, the value of a that satisfies all conditions is a = -1.
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