Equation of Tangents to the Circle x^2 + y^2 = 25 Making an Angle of 60 Degrees with the Positive Direction of the Y-Axis

To find the equation of tangents to the circle x^2 + y^2 = 25 that make an angle of 60 degrees with the positive direction of the y-axis, we can follow these steps:

Step 1: Find the Slope of the Tangent Line
The slope of the tangent line to a circle at a given point is equal to the negative reciprocal of the slope of the radius passing through that point. In this case, since the angle of 60 degrees is formed with the positive direction of the y-axis, we can say that the slope of the radius at the point of tangency is the negative reciprocal of the slope of the y-axis, which is 0.

Therefore, the slope of the tangent line is undefined.

Step 2: Identify the Point of Tangency
To find the point of tangency, we need to solve the system of equations formed by the circle equation and the equation of the tangent line. Let's assume the point of tangency is (a, b).

Substituting the coordinates (a, b) into the circle equation, we get:
a^2 + b^2 = 25

Step 3: Write the Equation of the Tangent Line
Using the point-slope form of a line, the equation of the tangent line passing through the point (a, b) with an undefined slope is given by:
x - a = 0

Step 4: Solve for the Intersection Point
To find the intersection point between the circle and the tangent line, we need to solve the system of equations formed by the circle equation and the equation of the tangent line. Substituting the equation of the tangent line into the circle equation, we get:
(a^2) + b^2 = 25
(a^2) = 25 - b^2

Substituting this back into the equation of the tangent line, we get:
x - a = 0
x = a

Step 5: Determine the Value of 'a' and 'b'
Since the equation of the tangent line is x - a = 0, it implies that x = a. Substituting this into the equation (a^2) = 25 - b^2, we get:
(a^2) = 25 - b^2
x^2 = 25 - b^2
a^2 = 25 - b^2
a = ±√(25 - b^2)

Step 6: Substitute the Value of 'a' and 'b' into the Equation of the Tangent Line
Since the equation of the tangent line is x - a = 0, we can substitute the value of 'a' into the equation to get:
x - √(25 - b^2) = 0

Conclusion:
The equation of the tangents to the circle x^2 + y^2 = 25 that make an angle of 60 degrees with the positive direction of the y-axis is given by:
x - √(25 - b^2) = 0
Therefore,