Problems On Ellipse

 Problems On Ellipse

ELLIPSE BY CONCENTRIC CIRCLE METHODE, RECTANGLE METHOD, OBLONG METHOD, ARCS OF CIRCLE METHOD, RHOMBUS METHOD AND BY DIRECTRIX-FOCUS METHOD.

 ELLIPSE BY CONCENTRIC CIRCLE METHODE...

 1 ] Draw ellipse by concentric circle method take, take major axis 100 mm and minor axis 70 mm long.

 steps: 

1. Draw both axes as perpendicular bisector of each other & give name their ends as shown.


2. Taking their intersecting points as a center, Draw two concentric circles considering both as respective diameters.


3. DIvide both circle in 12 equal parts & name as shown.


4. From all the point outer circle draw a vertical line downwards and upwards respectively.


5. From all the points in the inner circle draw a horizontal line to intersection those vertical lines.


6. mark all intersecting points properly as those are the points on the ellipse.


7. Join all these points along with the end of both axes in a smooth possible curve. it is required ellipse.

ELLIPSE BY RECTANGLE METHOD...

 2 ] Draw ellipse by rectangle method. take major axis 100 mm and minor axis 70 mm long.

 Steps:

1. Draw a rectangle taking a major axis as sides.


2. In this rectangle draw both axes as perpendicular bisectors of each other.


3. For construction, select the upper left part of the rectangle. Divide vertical small side and horizontal long side into the same number of equal parts. Here we divided into four parts.


4. Name those as shown...


5. Now join all the vertical points 1, 2, 3, 4 to the lower end of the minor axis.


6. Then extend the C1 line up to D1 and marks that point. Similarly extend C2, C3, C4 lines up to D2, D3, & D4 lines.


7. Marks all the points properly and join all along with ends A and D in a smooth possible curve. Do Similar construction in the right side parts. along with the lower half of the rectangle. Join all points in a smooth curve. it is required ellipse.

 ELLIPSE BY OBLONG METHOD...

 3 ] Draw ellipse by the Oblong Method. Draw a parallelogram of 100 mm and 70 mm long sides with the included angle of 75º inscribe ellipse in it.

* steps are same as to privious case ( RECTANGLE METHOD )

Only in place of ractangle, here is a parallelogram.

 steps:

1. Draw a parallelogram with an angle of 75º taking a major axis as sides.


2. In this parallelogram draw both axes as perpendicular bisectors of each other.


3. For construction, select the upper left part of the parallelogram. Divide vertical small side and horizontal long side into the same number of equal parts. Here we divided into four parts.


4. Name those as shown...


5. Now join all the vertical points 1, 2, 3, 4 to the lower end of the minor axis.


6. Then extend the C1 line up to D1 and marks that point. Similarly extend C2, C3, C4 lines up to D2, D3,   & D4 lines.


7. Marks all the points properly and join all along with ends A and D in a smooth possible curve. Do Similar construction in the right side parts. along with the lower half of the parallelogram Join all points in a smooth curve. it is required ellipse.

ELLIPSE BY ARCS OF CIRCLE METHOD...

As per the definition of an ellipse is a locus of point p moving in a plane such that the sum of its distance from two fixed points ( F1 & F2 ) remains constants and equals to the length of major axis AB.

Note

A1 + B1 = AB

 A2 + B2 = AB

4 ] Major axis AB & minor axis CD are 100 mm and 70 mm long respectively, Draw ellipse by arcs of the circle method.

 Steps:

1. Draw both axes as usual. Name the ends & intersecting point.


2. Taking AO distance I.e. half major axis, from C, mark F1 & F2 On AB. ( focus 1 and 2 )


3. On line F1-O taking any distance, mark points 1, 2, 3, & 4.


4. Taking f1 center, with distance A1 draw an arc above AB and taking F2 center, with b1 distance cut this arc. Name the point p1.


5. Repeat this step with the same centers but taking now A2 & B2 distance for drawing arcs. Name the point p2.


6. Similarly get all other p points. with the same steps positions of P can be located below AB.


7. join all points by the smooth curve to get an ellipse.

ELLIPSE BY RHOMBUS METHOD...

5 ] Draw rhombus of 100 mm & 70 mm long diagonals and inscribe an ellipse in it.

Steps:

1. Draw rhombus of given dimensions.


2. Mark midpoints of all sides & name Those A, B, C, & D.


3. Join these points to the ends of smaller diagonals.


4. Mark points 1, 2, 3, 4 as four centers.


5. Taking 1 as a center and 1A radius draw an arc AB.


6. Taking 2 as center draw an arc CD.


7. Similarly taking 3 & 4 as centers and 3D radius draw arcs DA & BC.

ELLIPSE BY DIRECTRIX-FOCUS METHOD...

6 ] Point F is 50 mm from a line AB. A point P is moving in a plane such that the ratio of its distances from F and line AB remains constant and equals to 2/3 draw locus of point p.

Steps: 

Here eccentricity = 2/3

1. Draw a vertical line AB and point F 50 mm from it.


2. Divide 50 mm distance in 5 parts.


3. Name 2nd part from F as V. It is 20 mm and 30 mm from F and AB line respectively. It is the first point giving a ratio of its distances from F and AB 2/3 i.e 20/30.


4. From more points giving the same ratio such as 30/45, 40/60, 50/75, etc.


5. Taking 45, 60, and 75 mm distance from line AB, draw three vertical lines to the right side of it.


6. Now with 30, 40, and 50 mm distance in compass cut these lines above and below, with F as a center.


7. Join these points through V in a smooth curve. This is required through V. it is an Ellipse.

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