Question

Length of tangent from (3,4) to x²+y² = 9?

Answer 1

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 = \(\sqrt {(x_2 - x_1)^2 + (y_2 - y_1)^2}\) 
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 = \(\sqrt {(0 - 3)^2 + (0 - 4)^2}\) 
d = \(\sqrt {9 + 16}\)
d = \(\sqrt {25}\)
d = 5 units
tangent_length = \(\sqrt {d^2 - r^2}\)
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 = \(\sqrt {5^2 - 3^2}\) 
tangent_length = \(\sqrt {25 - 9}\)
tangent_length = \(\sqrt {16}\)
tangent_length = 4 
Therefore, the length of the tangent from the point (3, 4) to the circle x² + y² = 9 is 4 units.