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
tangentlength = $\sqrt {d^2 - r^2}$
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:
tangentlength = $\sqrt {5^2 - 3^2}$
tangentlength = $\sqrt {25 - 9}$
tangentlength = $\sqrt {16}$
tangentlength = 4
Therefore, the length of the tangent from the point (3, 4) to the circle x2 + y2 = 9 is 4 units.