CONIC SECTIONS 1. Inscribe a parabola in the given rectangle, with its axis parallel to the side AB

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Damon Fields
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1 Inscribe a parabola in the given rectangle, with its parallel to the side AB A D B C Inscribe a parabola in the rectangle below, with its vertex located midway along the side PQ. P S Q R CONIC SECTIONS 1

2 The and focus of a parabola are shown. (b) Locate the vertex and draw a portion of the curve. Locate a point P on the curve 50mm from. ind the centre of curvature for this point. eccentricity line centre of curvature 90 P CONIC SECTIONS 2

3 The figure below shows the focus of a parabola and the direction of the. The position of a point P on the curve is also shown. Show how the and vertex are located and draw a portion of the curve to include point P. (b) Draw a tangent to the curve at point P. P CONIC SECTIONS 3

4 P is a point on a parabola, PT is a tangent to the curve and DD is the. (b) Locate the focus and draw a portion of the curve. Locate the centre of curvature for a point 50mm from the. 50 D P normal T D centre of curvature CONIC SECTIONS 4

5 Draw a parabola having as its focus and having P and Q as points on the curve. Draw the latus rectum. latus rectum P Q CONIC SECTIONS 5

6 is the focus of a parabola, P is a point on the curve and D is a point on the. (b) Draw a portion of the curve. Draw a tangent to the curve which shall be parallel to the line D. tangent chord drawn parallel to D line drawn through midpoint of chord parallel to P CONIC SECTIONS 6 D

7 The plan and elevation of a cone are shown below. The cone has been cut by the plane S-S as indicated. Complete the plan of the cone and determine the true shape of the cut surface. S 90 S CONIC SECTIONS 7

8 The diagram below shows a cone which has been cut by the plane L-L. Illustrate how the focal point, vertex, and eccentricity of the cut surface are established. eccentricity = 1 as the section plane L-L is parallel to the extreme generator of the cone vertex focus L focal sphere L CONIC SECTIONS 8

9 The and focus of an ellipse are shown. The eccentricity is 2/3. (b) Draw the complete ellipse. Draw a tangent to the curve at a point P, 100mm from the. 45 tangent 100 CONIC SECTIONS 9

10 P is a point on an ellipse whose and are shown. The eccentricity of the curve is 3/4. Locate a position for the focus and draw half the curve. (b) Construct a tangent to the curve from point Q. Q point of contact P CONIC SECTIONS 10

11 The of an ellipse as well as two points A and B on the curve are given. The eccentricity of the ellipse is Locate the focus and draw a portion of the curve which passes through the points A and B (b) Locate the centre of curvature for the point A. A arcs cross at B 4 equal divisions centre of curvature for point A CONIC SECTIONS 11

12 is one of the focal points of an ellipse and P is a point on the curve. The eccentricity of the ellipse is 0.75 (b) Draw half the curve. Draw a tangent to the curve which makes an angle of 45 degrees with the, showing clearly how the point of contact is obtained. Ecc. = 0.75 = 3/4 Tangent passes through points where 45 degree normals meet major auxiliary circle. 4 equal divisions P 3 equal divisions Point of contact CONIC SECTIONS 12

13 is the focus of an ellipse and P and Q are points on the curve. The eccentricity is (b) Draw a portion of the curve. Draw a tangent to the curve which shall be parallel to the line P. Major auxiliary circle Q Point of Contact P Chord // to P and bisected CONIC SECTIONS 13

14 is the focus of an ellipse, is a vertex and D is a point on the. (b) Draw half of the curve. Draw both directrices of the ellipse. D CONIC SECTIONS 14

15 1 and 2 are the focal points of an ellipse and AB is a tangent to the curve. (b) Draw a portion of the curve. ind the point of contact between the tangent AB and the curve. normals from 1 and 2 cross major auxiliary circle at tangent B p concentric circles method A 1 2 CONIC SECTIONS 15

16 Given are the elevation and plan of a cone which has been cut by the plane C-C as shown. Complete the plan of the cone. (b) Determine the true shape of the cut surface. C C CONIC SECTIONS 16

17 Given in the diagram below is a cone which has been cut by the plane S-S. Locate the focal points, vertices and directrices for the cut surface. S S CONIC SECTIONS 17

18 The and focus of a hyperbola are shown. The eccentricity is Draw a portion of the hyperbola. eccentricity line latus rectum CONIC SECTIONS 18

19 Construct a double hyperbola in the rectangles below. CONIC SECTIONS 19

20 The, and vertex of a hyperbola are shown. The eccentricity of the curve is 4/3. Locate the focus of the hyperbola and draw a portion of the curve. eccentricity line latus rectum CONIC SECTIONS 20

21 The, and a point P on the curve of a hyperbola are given. The eccentricity of the curve is 1.5. Locate the focus and vertex of the hyperbola. (b) Draw a portion of the curve to pass through point P. 2 units P Note: Ecc. = 1.5 = 2/3 R. 3 units CONIC SECTIONS 21

22 is a focal point of a hyperbola and P is a point on the curve. The eccentricity of the curve is Locate the and draw a portion of the curve to include point P. (b) Determine the centre of curvature for the point P. P Note: Ecc. is 1.25 = 5/4 P = 50mm = 5 units 5/4 = 50/40 D = 40mm = 4units CONIC SECTIONS 22 centre of curvature for point P

23 is the vertex of a hyperbola, is a focal point of the curve and D is a point on the. Draw a portion of the curve. D 45 CONIC SECTIONS 23

24 is one of the focal points of a double hyperbola, AP is a tangent to the curve and P is the point of contact. (b) Draw a portion of the curve. Locate the directrices and asymptotes of the curve. asymptotes directrices P = = A 90 Asymptotes and directrices meet at auxiliary circle CONIC SECTIONS 24

25 is one of the focal points of a double hyperbola, A is a point on one branch and B is a point on the other branch. The transverse is 50mm long. (b) Determine the position of the second focal point and draw a portion of the double curve. Determine the asymptotes to the curve. R85mm arc from A (35mm + 50mm) asymptotes R55mm arc from B (105mm - 50mm) position of 1 A B CONIC SECTIONS 25

DESIGN & COMMUNICATION GRAPHICS Conic Sections 1

DESIGN & COMMUNICATION GRAPHICS Conic Sections 1 The projections of a right cone are shown below. The traces of a simply inclined plane VTH are also given. The plane is parallel to an element of the cone. The intersection of a plane and a right cone