Circle

Step 1: Let the center of the circle be at and its radius be .
The circle touches the y-axis, so the distance from the center to the y-axis must be equal to the radius .
The distance from the point to the y-axis is simply , so we have:
Step 2: Next, the circle touches the line , so the distance from the center to this line must also be equal to the radius . The formula for the distance from a point to the line is given by:

For the line , we have , , and .
Thus, the distance from the center to the line is:

Since this distance must equal the radius , we have:
Step 3: Now, equating the two expressions for the radius , we get:
Squaring both sides:

Multiplying through by 2:
Expanding the right-hand side:
Simplifying:
Rearranging:
Thus, the locus of the center of the circle is given by:

Given: Point P with coordinates (3, 4) Equation of the circle: x² + y² = 9
The equation of the circle is in standard form (x - h)² + (y - k)² = r₂, where (h, k) represents the center of the circle. In this case, h = 0 and k = 0, so the center of the circle is at the origin (0, 0).
The distance (d) between two points (x₁, y₁) and (x₂, y₂) is given by:
d =
In this case, the coordinates of the point P are (3, 4), and the coordinates of the center of the circle are (0, 0). Thus, the distance between the point P and the center of the circle is:
d =
d =
d =
d = 5 units
tangent_length =
In this case, the radius (r) of the circle is 3 (since the equation of the circle is x² + y² = 9. Thus, the length of the tangent from point P to the circle is:
tangent_length =
tangent_length =
tangent_length =
tangent_length = 4
Therefore, the length of the tangent from the point (3, 4) to the circle x² + y² = 9 is 4 units.