Question

Prove that the line \( y=x+2 \) touches the circle \( x^{2}+y^{2}= \) 2. Find its point of contact.

Answer 1

y = x + 2 and x² + y² = 2

Then x² + (x+2)² = 2

Solving we get the solution is the point off contact is: (-1,1)

Answer 2

line: y=x+2

circle: x²+y²=2

x^2 + (x+2)^2 = 2

x^2 + x^2 + 4 + 4x = 2

2x^2 + 4x + 2 = 0

x^2 + 2x + 1 = 0

x^2 + x + x + 1 = 0

x(x+1)+1(x+1)

x = -1,-1

now put it in equation of line or either circle to get y=1.

hence point of contact becomes P(-1,1)