find b so that the line y = 2x + b is tangent to the circle x2 + y2 = 4

Question

find b so that the line y = 2x + b is tangent to the circle x2 + y2 = 4

Answer 1

1) y = 2*x + b

 

2) x² + y² = 4       Plugging equation 1) into equation 2):

 

x² + (2*x + b)² = 4*x² + 4*x2 + 4*x*b + b2 = 4         (arrange into quadratic form:)

 

A*x2 + B*x + C = 0          5*x² + (4*b)*x + (b² - 4) = 0       where: A = 5     B = (4*b)       C = (b² - 4)

 

quadratic solution for x to solve for y-intercept (at x = 0): x = [- B +/- √(B² - 4*A*C)]/2*A = 0

 

Note: To find y-intercept, b, Need to solve for b at x = 0

 

√(B² - 4*A*C) = 0    

 

B² = 4*A*C         A = 5       B = 4*b       C = b² - 4

 

16*b² = 4*(5)*(b² - 4)        

 

(divide all terms by 4)  

 

4*b² = 5*b² - 20      

 

b² = 20

 

b = √(20) = 2√(5)

 

 

Answer 2

y=2x+b

x^2+y^2=4

substitute

x^2+(2x+b)^2=4

x^2+4x^2+4xb+b^2=4

5x^2+4bx+(b^2-4)=0

The quadratic equation must have its discriminant equal to 0 in order for the line and the circle to be tangent.

Use the quadratic equation to find the discriminant, b^2-4ac.

b=4b, a=5, c=(b^2-4)

(4b)^2-(4)(5)(b^2-4)=0

16b^2-20b^2+80=0

-4b^2+80=0

4b^2=80

b^2=20

b=√20

b=2√5 (answer)

using y=2x+2√5 and x^2+y^2=4, you find that the point of intersection is (-4√5/5, 2√5/5)