Final answer:
To find the value of a for point P(a, -1) with a distance of 5 units from point Q(3, 2), the distance formula is used, leading to two potential solutions where a can be -1 or 7.
Step-by-step explanation:
If the distance between the point P(a, -1) and Q(3, 2) is 5 units, we can find the value of a using the distance formula for two points in the Cartesian coordinate system.
The distance formula is given by:
D = √[(x2 - x1)² + (y2 - y1)²], where (x1, y1) and (x2, y2) are the coordinates of the two points.
Here, we have:
P(a, -1) = (x1, y1)
Q(3, 2) = (x2, y2)
D = 5 units
Plugging the values into the distance formula, we get:
5 = √[(3 - a)² + (2 - (-1))²]
Squaring both sides to remove the square root, we obtain:
25 = (3 - a)² + 3²
Simplifying, we have:
25 = (3 - a)² + 9
Subtracting 9 from both sides, we get:
16 = (3 - a)²
Now, we take the square root of both sides:
4 = ±(3 - a)
So, a has two possible values from the two equations:
4 = 3 - a or -4 = 3 - a
Solving for a in each case:
a = 3 - 4 = -1
a = 3 + 4 = 7
Thus, the value of a can either be -1 or 7.