Question

The equations of the tangents drawn at the ends of the major axis of the ellipse 9x² + 5y² - 30y = 0 are

(a) y = ± 3
(b) x = ± √5
(c) y = 0, y = 6
(d) None of the above

Answer 1

The equations of the tangents to the ellipse at the endpoints of its major axis are y = 0 and y = 6, which corresponds to option (c).

Step-by-step explanation:

The student is asking for the equations of the tangents to an ellipse at the endpoints of its major axis. The given ellipse is 9x² + 5y² - 30y = 0. To find the major axis endpoints, we first need to complete the square for the y-term:

9x² + 5(y² - 6y + 9) - 45 = 0
to make it
9x² + 5(y - 3)² = 45

Then the ellipse in standard form is:

\(rac{x²}{5} + rac{(y - 3)²}{9} = 1\)

This shows the major axis is parallel to the x-axis and has the center (0, 3). Since the major axis endpoints lie on the y-axis (because x=0), we can determine their coordinates: (0,0) and (0,6). Therefore, the tangents at these points are horizontal lines with equations y = 0 and y = 6 respectively, which corresponds to option (c).