Find the points of intersection of the circle x² + y² = 1c line: y = 2x + 1 y = x + 1

1) Find the points of intersection of the circle x ^ 2 + y ^ 2 = 1 with the straight line y = 2x + 1.

To do this, we solve the system of equations:

x ^ 2 + y ^ 2 = 1;

y = 2x + 1.

Substituting into the first equation the value y = 2x + 1 from the second equation, we get:

x ^ 2 + (2x + 1) ^ 2 = 1;

x ^ 2 + 4x ^ 2 + 4x + 1 = 1;

5x ^ 2 + 4x + 1 = 1;

5x ^ 2 + 4x + 1 – 1 = 0;

5x ^ 2 + 4x = 0;

5x * (x + 4/5) = 0;

x1 = 0;

x2 = -4/5 = -0.8.

We find at:

y1 = 2×1 + 1 = 2 * 0 + 1 = 1;

y2 = 2×2 + 1 = 2 * (-0.8) + 1 = -1.6 + 1 = -0.6.

Answer: (0; 1) and (-0.8; -0.6).

2) Find the points of intersection of the circle x ^ 2 + y ^ 2 = 1 with the straight line y = x + 1.

To do this, we solve the system of equations:

x ^ 2 + y ^ 2 = 1;

y = x + 1.

Substituting into the first equation the value y = x + 1 from the second equation, we get:

x ^ 2 + (x + 1) ^ 2 = 1;

x ^ 2 + x ^ 2 + 2x + 1 = 1;

2x ^ 2 + 2x + 1 = 1;

2x ^ 2 + 2x + 1 – 1 = 0;

2x ^ 2 + 2x = 0;

2x * (x + 1) = 0;

x1 = 0;

x2 = -1.

We find at:

y1 = x1 + 1 = 0 + 1 = 1;

y2 = x2 + 1 = -1 + 1 = 0.

Answer: (0; 1) and (-1; 0).