$\left\{\begin{array}{l}x^2+y^2=2 b \\ 3 x+y=9\end{array}\right.
Answer (2)
Solve the second equation for y : y = 9 − 3 x .
Substitute into the first equation and simplify: 10 x 2 − 54 x + ( 81 − 2 b ) = 0 .
Calculate the discriminant: D = 80 b − 324 .
Set D ≥ 0 and solve for b : b ≥ 20 81 .
The system has real solutions when b ≥ 20 81 .
Explanation
Problem Analysis We are given the system of equations: x 2 + y 2 = 2 b 3 x + y = 9 Our goal is to find the value of b for which the system has real solutions.
Solving for y First, we solve the second equation for y in terms of x :
y = 9 − 3 x
Substitution and Simplification Next, we substitute this expression for y into the first equation: x 2 + ( 9 − 3 x ) 2 = 2 b Expanding and simplifying, we get: x 2 + ( 81 − 54 x + 9 x 2 ) = 2 b 10 x 2 − 54 x + 81 = 2 b 10 x 2 − 54 x + ( 81 − 2 b ) = 0
Calculating the Discriminant For the quadratic equation to have real solutions, the discriminant must be greater than or equal to zero. The discriminant is given by D = B 2 − 4 A C , where A = 10 , B = − 54 , and C = 81 − 2 b .
D = ( − 54 ) 2 − 4 ( 10 ) ( 81 − 2 b ) D = 2916 − 40 ( 81 − 2 b ) D = 2916 − 3240 + 80 b D = 80 b − 324
Solving for b We set the discriminant greater than or equal to zero: 80 b − 324 ≥ 0 Solving for b , we get: 80 b ≥ 324 b ≥ 80 324 b ≥ 20 81 b ≥ 4.05
Final Answer Therefore, the system has real solutions when b ≥ 20 81 or b ≥ 4.05 .
Examples
This problem demonstrates how the intersection of a circle and a line can be determined algebraically. In real life, this could model the path of an object (line) in relation to a circular area (e.g., a radar's range). By finding the values of 'b' for which the line and circle intersect, we determine when the object enters the radar's range. This has applications in tracking, navigation, and collision avoidance systems.
To find the values of b for which the equations x 2 + y 2 = 2 b and 3 x + y = 9 have real solutions, we derive that b must be greater than or equal to 20 81 . Therefore, the system has real solutions when b ≥ 4.05 . This stems from analyzing the discriminant of the resulting quadratic equation.
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