UNIT_I - Engineering Curves

UNIT_I

1. (a) Construct a rectangular hyperbola when a point P on it is at a distance of 18 mm and 34 mm from two asymptotes. Also draw a tangent to the curve at a point 20 mm from an asymptote.

(b) The vertex of a hyperbola is 60 from its focus. Draw the curve, if the eccentricity is 3/2. Draw a normal and a tangent at a point on the curve, 75 from the directrix.

2. (a) The major axis of an ellipse is 100 mm long and the distance between its foci is 70 mm. Draw the ellipse.

(b) Draw a hyperbola having the double ordinate of 100 mm, the abscissa of 60 mm and the transverse axis of 100 mm.

3. (a) Construct a pentagon length of a side is 30 mm.

(b) Draw an arc passing through three points not in straight line.

(c) Construct a parabola, with the distance of the focus from the directrix as 50 mm, also draw normal and tangent to the curve at a point 40 from the directrix.

4. (a) Inscribe a regular octagon in a given square of 50 mm side.

(b) Construct a parabola with base 60 and length of the axis 40. Draw a tangent to the curve at point 20 from the base. Also locate the focus and directrix to the parabola.

5. A fixed point is 75 mm from a fixed straight line. Draw the locus of a point P moving such a way that its distance from the fixed straight line is (i) twice its distance from the fixed point (ii)equal to its distance from the fixed point. Name the curves.

6. (a) Two points A&B are 100 mm apart. A point C is 75mm from A and 60mm from B. Draw the ellipse passing through A,B and C.

(b) A ball thrown up in the air reaches maximum height of 45 m and travels a horizontal distance of 75m. Trace the path of the ball, assuming it to be parabolic.

7. (a) Inscribe an ellipse in a parallelogram having sides 150 mm and 100 mm long and an included angle of 120⁰.

(b) A point P is 30 mm and 50 mm respectively from two straight lines which are at right angles to each other. Draw the rectangular hyperbola from P within 10 mm distance from each line.

8. (a) A parallelogram has sides 100 &80 mm at an included angle of 700. Inscribe an ellipse in the parallelogram. Find the major and minor axis of the curve.

(b) Draw an ellipse by concentric circles method and find the length of the minor axis with the help of the following data: (i) major axis 100 mm. (ii) distance between foci 80 mm.

9. (a) The major and minor axis of an ellipse is 120&80 mm. Draw an ellipse by arcs of circles method.

(b) The asymptotes of a hyperbola are inclined at 70⁰ to each other. Construct the curve when a point p on it is at a distance of 20 mm and 30 mm from the two asymptotes.

10. Two fixed points A&B are 100 mm apart. Trace the complete path of a point P moving (in the same plane as that of A&B) in such a way that, the sum of its distances from A&B is always the same and equal to 125 mm. Name the curve. Draw another curve parallel to and 25 mm away from this curve.

11. The foci of an ellipse are 90 mm apart and the minor axis is 65 mm long. Determine the length of the major axis and draw half the ellipse by concentric-circles method and other half by oblong method.

12. A circle of 60 mm diameter rolls without slipping on the outside of another circle of diameter 150 mm. Show the path of a point on the periphery of the (generating) rolling circle, diametrically opposite to the initial point of contact between the circle.

13. A circle of 60 mm diameter rolls on a horizontal line for a half revolution and then on a vertical line for another half revolution. Draw the curve traced out by a point p on the circumference of the circle.

14. Draw one turn of the involutes of a hexagon whose inscribed circle is 30mm in diameters.

15. Draw a straight line AB of any length. Make a point F, 65mm from AB .Trace the paths of a point P moving in such a way that the ratio of its distance from the point F, to its distance from AB is (a) 1(b) 2:3 Name each curve. Draw a normal and a tangent to each curve at a point on it 45 mm from F.

16. A fixed point is 75mm from a fixed straight line. Draw the locus of a point P moving such a way that its distance from the fixed straight line is equal to its distance from the fixed point. Name the curve. Draw a normal and tangent on the curve.

17. A fixed point F is 7.5 cm from a fixed straight line. Draw the locus of a point P moving in such a way that its distance from the fixed straight line is 2/3 times its distance from F. Name the curve. Draw normal and tangent at a point 6 cm from F.

18. Two fixed points A and B are 100 mm apart. Trace the complete path of a point P moving (in the same plane as that of A and B) in such a way that the sum of its distances from A and B is always equal to 125 mm. Name the curve. Draw another curve parallel to and 25 mm away from this curve.

19. The major axis of an ellipse is 150mm long and the minor axis is 100mm long. Find the foci and draw an ellipse by ‘arcs of circles’ method. Draw a tangent to the ellipse at a point on it 25mm above the major axis.

20. The vertex of a hyperbola is 65mm from its focus. Draw the curve if the eccentricity is 3/2. Draw a normal and a tangent at a point on the curve, 75 mm from the directrix.

21. Show by means of a drawing that when the diameter of the directing circle is twice that of the generating circle, the hypocycloid is a straight line. Take the diameter of the generating circle equal to 50mm.

22. The foci of an ellipse are 80mm apart and the minor axis is 55mm long. Determine the length of the major axis and draw the ellipse by concentric-circle method. Draw a curve parallel to the ellipse and 20mm away from it.

23. A circle of 50mm diameter rolls on the circumference of another circle of 175mm diameter and outside it. Trace the locus of a point on the circumference of the rolling circle for one complete revolution. Name the curve. Draw a tangent and a normal to the curve at a point 125mm from the center of the directing circle.

24. Two straight lines OA and OB make an angle of 75⁰ between them. P is a point 40mm from OA and 50mm from OB. Draw a hyperbola through P, with OA and OB as asymptotes, marking at least ten points.

25. A circle of 35mm diameter rolls on a horizontal line. Draw the curve traced out by a point R on the circumference for one half revolution of the circle. For the remaining half revolution, the circle rolls on the vertical line. The point R vertically above the center of the circle in the starting position.

26. (a) Inscribe an ellipse in a parallelogram having sides 150mm and 100mm long and an inclined angle of 120⁰.

(b) Draw a rectangle having its sides 125mm and 75mm long. Inscribe two parabolas in it with their axis bisecting each other.

27. Draw a hypo cycloid of a circle of 30mm diameter which rolls inside another circle of 160mm diameter, for one revolution counter clock wise. Draw a tangent and a normal to it at a point 60mm from the center of the directing circle.

28. Two points A and B are 50 mm apart. Draw the curve traced out by a point P moving in such a way that the difference between its distances from A and B is always constant and equal to 20 mm.

29. Show by means of a drawing that when the diameter of the directing circle is twice that of the generating circle, the hypocycloid is a straight line. Take diameter of generating circle equal to 60 mm.

30. The major axis of an ellipse is 150 mm long and the minor axis is 100 mm long. Find the foci and draw the ellipse by ‘arcs of circles’ method. Draw a tangent to the ellipse at a point on it 25 mm above the major axis.

31. A point P is 30 mm and 50 mm respectively from two straight lines which are at right angles to each other. Draw a rectangular hyperbola from P within 10 mm distance from each line.

32. A fixed point is 75mm from a fixed straight line. Draw the locus of a point P moving such a way that its distance from the fixed straight line is (a) twice its distance from the fixed point (b) equal to its distance from the fixed point. Name the curves.

33. A circle of 40mm diameter rolls on a straight line without slipping. In the initial position the diameter PQ of the circle is parallel to the line on which it rolls. Draw the locus of the points P and Q for one complete revolution of the circle.

34. Draw a rectangle having its sides 125mm and 75mm long. Inscribe two parabolas in it with their axis bisecting each other.

35. The foci of an ellipse are 90mm apart and the minor axis is 65mm long. Determine the length of the major axis and draw half of the ellipse by concentric circle method and other half by oblong method. Draw a curve parallel to the ellipse and 25mm away from it.

36. Draw a hypocycloid of a circle of 45mm diameter which rolls inside another circle of 200 mm diameter for one revolution. Draw a tangent and normal at any point on it.

37. A circle of 115 mm diameter rolls on another circle of 75mm diameter with internal contact. Draw the locus of a point on the circumference of rolling circle for its one complete revolution.

38. Draw a hypocycloid of a circle of 45mm diameter which rolls inside another circle of 200 mm diameter for one revolution. Draw a tangent and normal at any point on it.

39. A coin of 40mm diameter rolls over horizontal table without slipping. A point on the circumference of the coin is in contact with the table surface in the beginning and after one complete revolution. Draw and name the curve. Draw a tangent and normal at any point on the curve.

40. A circle of 50mm diameter rolls along a straight line without slipping. Draw the curve traced out by a point P on the circumference for 1.5 revolution of the circle. Name the curve. Draw a tangent and normal at a point on it 35mm from the line.

41. (a) A fountain jet discharges water from ground level at an inclination of 50⁰ to the ground. The jet travels a horizontal distance of 9 m from the point of discharge and falls on the ground. Trace the path of the jet.

(b) The distance between two fixed points is 90mm. A point P moves such that the difference of its distance from the two fixed points is always equal to 60mm.Draw the loci of P.

42. Two fixed points A and B are 100mm apart. Trace the complete path of a point P moving (in the same plane as that of A and B) in such a way that, the sum of its distances from A and B is always the same and equal to 125 mm. Name the curve. Draw another curve parallel to and 25mm away from this curve.

43. A circle of 50mm diameter, rolls on a horizontal line for half a revolution clock wise and then on a line inclined at 60⁰ to the horizontal for another half clockwise. Draw the curve traced by a point P on the circumference of the circle, taking the top most point on the rolling circle as the initial position of the generating point.