Question

At which points does the graph of the equation `25x^2+16y^2+200x-160y+400=0` have a vertical or horizontal tangent line?

At which points does the graph of the equation `25x^2+16y^2+200x-160y+400=0` have a vertical or horizontal tangent line?

Answer 1

Horizontal at (4, 0) and (4, 10) and vertical at (0, 5) and (8, 5)>

Explanation:

The equation can be reorganized to the standard form

`((x-4)/4)^2+((y-5)/5)^2=1`

that represents an ellipse with center C(4, 5), semi axes minor a = 4

and major b = 5 and axes parallel to the axes of coordinates..

The ends of the axes of the ellipse are A(4, 10), A'(4, 0) B(8, 5) and

B'(0, 5).

At A and A', the tangents y = 10 and y = 0 are parallel to x-axis.

At B and B', the tangents x = 8 and x = 0 are parallel to y-axis.