HOW TO SHOW A LINE DOES NOT INTERSECT A CIRCLE

Problem 1 :

Show that the line y = 2x + 1 does not intersect the circle with equation x² + y² - 2x + 4y + 1 = 0

Solution :

y = 2x + 1 -----(1)

x² + y² - 2x + 4y + 1 = 0 -------(2)

Applying the value of y in (2), we get

x² + (2x + 1)² - 2x + 4(2x + 1) + 1 = 0

x² + 4x² + 4x + 1 - 2x + 8x + 4 + 1 = 0

5x² + 10x + 6 = 0

Finding the nature of this quadratic equation, we get

a = 5, b = 10 and c = 6

Discriminant = b² - 4ac

= 10² - 4(5) (6)

= 100 - 120

= -20 < 0

The roots are not real, so the circle and line does not touch each other.

Problem 2 :

Show that the line y = 2x + 1 is a tangent to the circle

x² + y² - 6x - 4y + 8 = 0

Solution :

y = 2x + 1  ----(1)

x² + y² - 6x - 4y + 8 = 0 ----(2)

Applying the value of y in (2), we get

x² + (2x + 1)² - 6x - 4(2x + 1) + 8 = 0

x² + 4x² + 4x + 1 - 6x - 8x - 4 + 8 = 0

5x² - 10x + 5 = 0

x² - 2x + 1 = 0

Discriminant = b² - 4ac

a = 1, b = -2 and c = 1

= (-2)² - 4(1) (1)

= 4 - 4

= 0

So, the line is intersecting the circle.