At what points will the line y = x intersect the unit circle x² + y² = 1?

Solution:

The two equations given are

y = x …. (1)

x² + y² = 1 …. (2)

Substitute equation (1) in (2)

x² + x² = 1

2x² = 1

x² = ½

x = √1/2

x = ±0.707

As the square can be positive or negative we get x1 = 0.707 and x2 = -0.707

From equation (1) we know that y = x

y₁= x₁= 0.707

y₂= x₂ = -0.707

Therefore, the line y = x intersects the unit circle x² + y² = 1 at (0.707, 0.707) and (-0.707, -0.707).

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At what points will the line y = x intersect the unit circle x² + y² = 1?

Summary:

The line y = x intersects the unit circle x² + y² = 1 at (0.707, 0.707) and (-0.707, -0.707).