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To determine the eccentricity of the ellipse, we can use the formula:

e = sqrt(1 - (a^2/b^2))

where a is the distance between the center and the vertex, and b is the distance between the center and the minor axis. In this case, a = 150mm and b = 100mm (since the distance between the directrix and vertex is 150mm, and the distance between the directrix and minor axis is 50mm). Substituting these values into the formula, we get:

e = sqrt(1 - (150^2/100^2)) = sqrt(1 - 2.25) = sqrt(-1.25)

Since the value under the square root is negative, this means that the eccentricity is imaginary, which implies that there is no real solution for this ellipse.

To construct an ellipse using the directrix-focus method, we can follow these steps:

1. Draw a line segment AB of length equal to the major axis of the ellipse (in this case, it is 2a = 300mm).

2. Bisect AB at point O, and draw a perpendicular line OC through O.

3. Mark a point F on OC such that OF = a*e, where e is the eccentricity.

4. Draw a line through F perpendicular to OC, and extend it to intersect AB at points P and Q.

5. Draw lines from P and Q to the two given directrices (which are parallel lines on either side of OC).

6. The intersection points of these lines with AB will be the endpoints of the major axis of the ellipse.

7. Draw a perpendicular line through O to intersect the major axis at its midpoint M.

8. Draw a line through M parallel to OC, and extend it to intersect the directrices at points S and T.

9. The distance between S and T will be equal to the length of the minor axis of the ellipse (in this case, it is 2b = 200mm).

10. Draw an arc with center at point O, passing through points S and T, to complete the ellipse.

To draw the tangent and normal to the curve at any point S on the ellipse, we can follow these steps:

1. Draw a line from point S to the center of the ellipse.

2. Draw a perpendicular line from point S to the major axis of the ellipse.

3. The intersection point of these two lines will be the point of tangency T.

4. Draw a line from point T to point S to represent the tangent to the curve at point S.

5. Draw a line from point S to the center of the ellipse to represent the radius vector.

6. Draw a perpendicular line to this radius vector at point S to represent the normal to the curve at point S.

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To determine the eccentricity of the ellipse, we can use the formula:

e = sqrt(1 - (150^2/100^2)) = sqrt(1 - 2.25) = sqrt(-1.25)

Since the value under the square root is negative, this means that the eccentricity is imaginary, which implies that there is no real solution for this ellipse.

To construct an ellipse using the directrix-focus method, we can follow these steps:

1. Draw a line segment AB of length equal to the major axis of the ellipse (in this case, it is 2a = 300mm).

2. Bisect AB at point O, and draw a perpendicular line OC through O.

3. Mark a point F on OC such that OF = a*e, where e is the eccentricity.

4. Draw a line through F perpendicular to OC, and extend it to intersect AB at points P and Q.

5. Draw lines from P and Q to the two given directrices (which are parallel lines on either side of OC).

6. The intersection points of these lines with AB will be the endpoints of the major axis of the ellipse.

7. Draw a perpendicular line through O to intersect the major axis at its midpoint M.

8. Draw a line through M parallel to OC, and extend it to intersect the directrices at points S and T.

9. The distance between S and T will be equal to the length of the minor axis of the ellipse (in this case, it is 2b = 200mm).

10. Draw an arc with center at point O, passing through points S and T, to complete the ellipse.

To draw the tangent and normal to the curve at any point S on the ellipse, we can follow these steps:

1. Draw a line from point S to the center of the ellipse.

2. Draw a perpendicular line from point S to the major axis of the ellipse.

3. The intersection point of these two lines will be the point of tangency T.

4. Draw a line from point T to point S to represent the tangent to the curve at point S.

5. Draw a line from point S to the center of the ellipse to represent the radius vector.

6. Draw a perpendicular line to this radius vector at point S to represent the normal to the curve at point S.

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