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What is the equation of the tangent to the circle [tex]x^2 + y^2 = 25[/tex] that is inclined at 60 degrees with the x-axis?

Answer :

Final answer:

The equation of the tangent to the circle x² + y² = 25 that is inclined at 60 degrees with the x-axis is y = (√3)x - 5√3.

Explanation:

The equation of the tangent to the circle x² + y² = 25 that is inclined at 60 degrees with the x-axis is y = (√3)x - 5√3.To find the equation of the tangent to the circle x² + y² = 25 that is inclined at 60 degrees with the x-axis, we can first find the slope of the tangent line.

The slope of a line inclined at 60 degrees with the x-axis is equal to the tangent of 60 degrees, which is √3. Now, we can use the point-slope form of a line to determine the equation of the tangent line.Since the circle x² + y² = 25 is centered at the origin, the point (x, y) on the circle that corresponds to the tangent point will also be the point of tangency.

Let's denote this point as (x₁, y₁). We can substitute this point and the slope (√3) into the point-slope form of a line to find the equation of the tangent line: y - y₁ = (√3)(x - x₁). Since the circle has a radius of 5, we can choose any point on the circle and substitute its coordinates into the equation. Let's choose the point (5, 0): y - 0 = (√3)(x - 5) => y = (√3)x - 5√3.

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