Solve for .
I get that the slopes of the line for x = 1 are and .
For the -1 slope, the slope of the normal line is +1. I get the line y = x. (See the graph below. This point and line is green.)
For the 0 slope, the slope of the normal is undefined. I get the line x = 1. (This is the blue point and line. Never mind that the blue line also intersects the green point. That's just an artifact of this normal line being parallel to the y-axis.
You know the equation of the normal line, thus you know the slope of the normal line. (Let's just consider the y= x normal line for simplicity.) So you are looking for a point on the ellipse where the normal line has a slope of 1. Thus the tangent line has a slope of -1. Thus:
Solve this equation for y and you have the equation of the tangent line that intersects the ellipse where the normal line has a slope of -1. I get the line y = x.
Now you want to find the points where this line intersects the ellipse. So plug y = x into your ellipse formula and solve for x. (There will be typically be two x values also, one of which you already know.) Then you can find the y value for these points by plugging x into y = x. Generally the two points will not be on the same line, as they are in this case.