Find the equation of an ellipse if its center is S(2,1) and the edges of a triangle PQR are tangent lines to this ellipse. P(0,0), Q(5,0), R(0,4)

Question

Find the equation of an ellipse if its center is S(2,1) and the edges of a triangle PQR are tangent lines to this ellipse. P(0,0), Q(5,0), R(0,4)

Answer 1

First what you need to do - is draw the picture. If you label the vertices of a Δ PQR according to the given coordinates, it is easy to see that this triangle is a right triangle with angle P as a right angle. Make sure to sketch the ellipse so, that you see its center S has coordinate x = 2 and y = 1.

As we know the standard form of ellipse equation is (x - h)² / a² + (y - k)² / b² = 1. The coordinates of the center S of the ellipse are given. Thus, we can find that h = 2 and k = 1 units. To find the values of a - the length of major axis, and b - the length of minor axis, let's look at the picture attentively.

From the picture you see that the distance from point S to the point, where side PQ is tangent to ellipse is 1 unit. It means that minor semiaxis has a length of 1 unit. The distance from the center of the ellipse to the point where side PR is tangent to the ellipse is 2 units in length. It means that the major semiaxes of the ellipse is 2 units. Knowing these facts, we can write:

1. a = 2 x 2 = 4 units
2. b = 2 x 1 = 2 units

Putting all values to the standard form of ellipse equation we get:

(x - 2)² / 4² + (y - 1)² / 2² = 1

Finally,

(x - 2)² / 16 + (y - 1)² / 4 = 1

P. S. Emily W. brought to my attention some interesting thoughts. When I was doing the problem, I have assumed that the major axis is parallel to x-axis and it is the standard problem. Now l see that was wrong and that the axis is tilted. I will think about this option and soon update the solution.

A tip for the students: please provide us the information about the level of the course you are talking.