Question

prove algebraically that the straight line with the equation x=2y+5 is a tangent of the circle with the equation x^2+y^2=5

Answer 1

Explanation:

To prove the straight line is tangent to the circle:

1. substitute the straight line into the circle equation
2. discriminant of the new equation has to be equals to 0

(1)

substitute x = 2y + 5 into x² + y² = 5

x² + y² = 5

(2y + 5)² + y² = 5

4y² + 20y + 25 + y² = 5

5y² + 20y + 20 = 0

y² + 4y + 4 = 0

(2)

discriminant of y² + 4y + 4 = 0 has to be equals to 0

`\boxed{D=b^2-4ac}`

D = 4² - 4(1)(4)

D = 0

∴ Proven