Let a, b and c be real numbers such that
a^2 + ab + b^2 = 9
b^2 + bc + c^2 = 52
c^2 + ca + a^2 = 49.
Compute the value of (49b^2+39bc+9c^2)/a^2.
To find the value of (49b^2 + 39bc + 9c^2) / a^2, we need to simplify the expression using the given equations.
Let's start by rearranging the equations:
1) a^2 + ab + b^2 = 9
2) b^2 + bc + c^2 = 52
3) c^2 + ca + a^2 = 49
Now, let's focus on the expression (49b^2 + 39bc + 9c^2) / a^2 and simplify it step by step:
Step 1: Divide the numerator and denominator by 9.
(49b^2 + 39bc + 9c^2) / a^2 = (7b^2 + 13bc + 3c^2) / (a^2 / 9)
Step 2: Substitute the values of a^2, b^2, and c^2 from the given equations:
(7b^2 + 13bc + 3c^2) / (a^2 / 9) = (7(9 - ab) + 13bc + 3(49 - ca)) / (a^2 / 9)
= (63 - 7ab + 13bc + 147 - 3ca) / (a^2 / 9)
= (210 - 7ab + 13bc - 3ca) / (a^2 / 9)
Step 3: Substitute the value of a^2 from the first equation:
(210 - 7ab + 13bc - 3ca) / (a^2 / 9) = (210 - 7ab + 13bc - 3ca) / (9 / 9)
= 210 - 7ab + 13bc - 3ca
Therefore, the simplified expression is 210 - 7ab + 13bc - 3ca.
To find the value of the expression (49b^2 + 39bc + 9c^2) / a^2, we need to find the values of a, b, and c that satisfy the given system of equations:
a^2 + ab + b^2 = 9 ---(1)
b^2 + bc + c^2 = 52 ---(2)
c^2 + ca + a^2 = 49 ---(3)
To simplify the problem, let's consider each equation and solve for one variable in terms of the others.
From equation (1):
a^2 + ab + b^2 = 9
Rearranging, we get:
a^2 + 2ab + b^2 - ab = 9
(a + b)^2 - ab = 9
Since we need to find (49b^2 + 39bc + 9c^2) / a^2, let's try to express the expression in terms of (a + b) and (c).
From equation (2):
b^2 + bc + c^2 = 52
Multiplying equation (2) by 4, we get:
4b^2 + 4bc + 4c^2 = 208
From equation (3):
c^2 + ca + a^2 = 49
Multiplying equation (3) by 4, we get:
4c^2 + 4ca + 4a^2 = 196
Adding the equations (2) and (3), we have:
4b^2 + 4c^2 + 4bc + 4c^2 + 4ca + 4a^2 = 208 + 196
8b^2 + 4bc + 8c^2 + 4ca + 4a^2 = 404
2(4b^2 + 2bc + 4c^2 + 2ca + 2a^2) = 404
Now, we can substitute the expression (a + b)^2 - ab = 9 into the equation above:
2((a + b)^2 - ab) = 404
2(a^2 + 2ab + b^2 - ab) = 404
2(a^2 + ab + b^2) = 404
2(9) = 404
18 = 404
This is a contradiction. Since there are no real numbers a, b, and c that satisfy the given system of equations, we cannot compute the value of the expression (49b^2 + 39bc + 9c^2) / a^2.