The problem has two related parts

1) let a be an arbitrary real number. Find the distance between the point (1,2) and the point (4,a). (answer should be an algebraic expression in terms of a).

2) find the value of b for which the point (0,b) is equidistant from (4,4) and (5,-4) (hint: find two different distances and set those distances equal

Why did you not just follow the instructions?

All you need is the distance formula between 2 points.

distance between the 2 points
= √((4-1)^2 + (a-2)^2 )
= √(a^2 - 4a + 13)

for #2, just set the two square roots equal to each other.
Simplify like I did above, then square both sides and solve for b
Let me know what you get

To solve both parts of this problem, we need to use the distance formula. The distance formula calculates the distance between two points in a coordinate plane.

1) Finding the distance between the point (1,2) and (4,a):
The distance formula is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)

Substituting the coordinates (1,2) and (4,a) into the formula:
d = √((4 - 1)^2 + (a - 2)^2)
= √(3^2 + (a - 2)^2)
= √(9 + (a - 2)^2)
= √(9 + a^2 - 4a + 4)
= √(a^2 - 4a + 13)

So, the distance between the points (1,2) and (4,a) is given by the expression √(a^2 - 4a + 13).

2) Finding the value of b when the point (0,b) is equidistant from (4,4) and (5,-4):
Let's calculate the distances from (0,b) to (4,4) and (5,-4), then set them equal to each other:

Using the distance formula:
Distance from (0,b) to (4,4):
d1 = √((4 - 0)^2 + (4 - b)^2)
= √(16 + (4 - b)^2)
= √(16 + (4 - b)^2)

Distance from (0,b) to (5,-4):
d2 = √((5 - 0)^2 + (-4 - b)^2)
= √(25 + (-4 - b)^2)
= √(25 + (-4 - b)^2)

Setting the two distances equal:
√(16 + (4 - b)^2) = √(25 + (-4 - b)^2)

Squaring both sides to eliminate the square root:
16 + (4 - b)^2 = 25 + (-4 - b)^2

Expanding the squares and simplifying:
16 + 16 - 8b + b^2 = 25 + 16 + 8b + b^2

Rearranging and canceling out terms:
32 - 8b = 41 + 8b

Bringing like terms to one side:
16b = 9

Solving for b:
b = 9/16

So, the value of b for which the point (0,b) is equidistant from (4,4) and (5,-4) is 9/16.