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?

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?
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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) = 0If the line is a tangent to the circle, this has a repeatedroot. 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√2This can now be substituted back into the meeting equationabove: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)
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