The value of y that minimizes the sum of the two distances from (3,5) to (1,y) and from (1,y) to (4,9) can be written as \frac{a}{b} where a and b are coprime positive integers. Find a + b.
To find the value of y that minimizes the given sum of distances, we need to minimize each individual distance and then find the value of y that achieves this minimum.
Let's start by finding the first distance from (3,5) to (1,y). This can be calculated using the distance formula:
d1 = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Substituting the coordinates (3,5) and (1,y):
d1 = sqrt((1 - 3)^2 + (y - 5)^2)
= sqrt((-2)^2 + (y - 5)^2)
= sqrt(4 + (y - 5)^2)
= sqrt(4 + y^2 - 10y + 25)
= sqrt(y^2 - 10y + 29)
Next, let's find the second distance from (1,y) to (4,9) using the same distance formula:
d2 = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Substituting the coordinates (1,y) and (4,9):
d2 = sqrt((4 - 1)^2 + (9 - y)^2)
= sqrt((3)^2 + (9 - y)^2)
= sqrt(9 + (9 - y)^2)
= sqrt(9 + y^2 - 18y + 81)
= sqrt(y^2 - 18y + 90)
Now, the sum of the two distances is given as:
S = d1 + d2
Plugging in the expressions we derived for d1 and d2:
S = sqrt(y^2 - 10y + 29) + sqrt(y^2 - 18y + 90)
Our goal is to find the value of y that minimizes this sum, which means we need to find the value of y that minimizes the function S. To do this, we can take the derivative of S with respect to y and set it equal to zero, and then solve for y.
dS/dy = (1/2) * (2y - 10) / sqrt(y^2 - 10y + 29) + (1/2) * (2y - 18) / sqrt(y^2 - 18y + 90)
Setting dS/dy = 0:
(1/2) * (2y - 10) / sqrt(y^2 - 10y + 29) + (1/2) * (2y - 18) / sqrt(y^2 - 18y + 90) = 0
Simplifying the equation:
(2y - 10) / sqrt(y^2 - 10y + 29) = (18 - 2y) / sqrt(y^2 - 18y + 90)
Then, cross-multiplying:
(2y - 10) * sqrt(y^2 - 18y + 90) = (18 - 2y) * sqrt(y^2 - 10y + 29)
Squaring both sides of the equation to eliminate the square roots:
(4y^2 - 40y + 100) * (y^2 - 18y + 90) = (324 - 36y + 4y^2) * (y^2 - 10y + 29)
Expanding and simplifying:
4y^4 - 40y^3 + 100y^2 - 72y^3 + 720y^2 - 1800y + 360y^2 - 3600y + 9000 = 324y^2 - 3600y + 1161 - 36y^2 + 408y - 1161
Combining like terms and canceling out common terms:
4y^4 - 112y^3 + 220y^2 - 5400y + 9000 = 0
Now, we have a quartic equation in y. To solve this equation, we can try factoring, but it might not be a straightforward process. Alternatively, we can use numerical methods to approximate the roots.