Answer:
y = -1/√3 (x − 4)
Step-by-step explanation:
Let's say the point where the tangent line intersects the circle is B (bₓ, bᵧ). Therefore:
bₓ² + bᵧ² = 4
The slope of the tangent line at that point is dy/dx, which we can find with implicit differentiation:
x² + y² = 4
2x + 2y dy/dx = 0
2y dy/dx = -2x
dy/dx = -x/y
At point B, the slope of the tangent line is m = -bₓ/bᵧ.
We can also write that as the slope between the points A(4, 0) and B(bₓ, bᵧ).
m = (bᵧ − 0) / (bₓ − 4)
m = bᵧ / (bₓ − 4)
Setting the expressions equal:
-bₓ / bᵧ = bᵧ / (bₓ − 4)
bᵧ² = -bₓ (bₓ − 4)
bᵧ² = 4 bₓ − bₓ²
Substituting this into the first equation:
bₓ² + (4 bₓ − bₓ²) = 4
4 bₓ = 4
bₓ = 1
Therefore, bᵧ = √3. So the slope of the line is -1/√3. The equation of the line is:
y − 0 = -1/√3 (x − 4)
y = -1/√3 (x − 4)
Graph: desmos.com/calculator/sornb1ajgp
