What is the point of contact when the tangent line y equals x plus c touches the circle x squared plus y squared equals 4?

Answer 1

The line meets the circle when:

y = x + c

→ x² + y² = 4

→ x² + (x + c)² - 4 = 0

→ x² + x² +2cx + c² - 4 = 0

→ 2x² + 2cx + (c² - 4) = 0

If the line is a tangent to the circle, this has a repeated root. This occurs when the discriminant is 0, ie when:

(2c)² - 4 × 2 × (c² - 4) = 0

→ 4c² - 8c² + 32 = 0

→ 4c² = 32

→ c² = 8

→ c = √8 = 2√2

This can now be substituted back into the meeting equation above:

2x² + 2cx + (c² - 4) = 0

→ 2x² + 2 × 2√2 × x + (√8)² - 4 = 0

→ 2x² + 4√2 x + 8 - 4 = 0

→ x² + 2√2 x + 2 = 0

→ (x + √2)² = 0

→ x = -√2

→ y = x + 2√2

= -√2 + 2√2

= √2

→ point of contact is (-√2, √2)