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
how are we suppose to help you without a graph
there is not graph provided
To solve these two problems, we'll need to calculate the distance between two points in each case.
1) Distance between (1,2) and (4,a):
To find the distance between two points (x1, y1) and (x2, y2), we use the distance formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
In our case, the coordinates of the two points are (1,2) and (4,a). Plugging these values into the distance formula, we get:
Distance = sqrt((4 - 1)^2 + (a - 2)^2)
Simplifying this expression gives us the distance between the points (1,2) and (4,a):
Distance = sqrt(9 + (a - 2)^2)
Therefore, the algebraic expression for the distance is sqrt(9 + (a - 2)^2).
2) Equidistant point with (0,b) from (4,4) and (5,-4):
To find the equidistant point, we need to calculate the distances between (0,b) and (4,4), and between (0,b) and (5,-4).
Again, using the distance formula, we have:
Distance1 = sqrt((4 - 0)^2 + (4 - b)^2)
Distance2 = sqrt((5 - 0)^2 + (-4 - b)^2)
Since the point (0,b) is equidistant from (4,4) and (5,-4), we can set the two distances equal to each other:
sqrt((4 - 0)^2 + (4 - b)^2) = sqrt((5 - 0)^2 + (-4 - b)^2)
Squaring both sides of the equation, we get:
(4 - 0)^2 + (4 - b)^2 = (5 - 0)^2 + (-4 - b)^2
Expanding and simplifying this equation gives:
16 + (4 - 2b + b^2) = 25 + (16 + 8b + b^2)
Combining like terms and canceling out common terms, we get:
2b = 25 - 16 + 8b
Re-arranging this equation yields:
-6b = 9
Dividing both sides by -6, we find:
b = -9/6 = -3/2
Therefore, the value of b for which the point (0,b) is equidistant from (4,4) and (5,-4) is -3/2.