JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech I Year ENGINEERING GRAPHICS Previous Question Paper (September - 2023 - R16)

An epicycloid is the locus of a point on the circumference of a circle (generating circle) that rolls without slipping on the outside of another fixed circle (directing circle). Given: Diameter of generating circle (d) = 45 mm (radius r = 22.5 mm); Diameter of directing circle (D) = 135 mm (radius R = 67.5 mm).

Steps for construction:

1. Calculate the angle subtended by the generating circle at the center of the directing circle: θ = (r / R) × 360° = (22.5 / 67.5) × 360° = 120°.
2. Draw the directing circle with radius R = 67.5 mm and mark the center O. Draw the initial vertical radius OA.
3. Draw the generating circle of radius r = 22.5 mm such that it is tangent to the directing circle at point P.
4. Divide the generating circle into 12 equal parts and the arc of 120° on the directing circle into 12 equal parts.
5. Draw the path of the center of the rolling circle (an arc of radius R + r = 90 mm).
6. Project the points from the generating circle onto the corresponding arcs drawn from center O to locate the points on the epicycloid curve. Join these points with a smooth curve.

Normal and Tangent construction:

1. Mark point M on the curve at a height of 10 mm above the center of the rolling circle position.
2. To draw the normal, join M to the center of the rolling circle at that specific position (let this be C). Extend this line to intersect the directing circle at point N.
3. The line MN is the normal to the curve at point M.
4. Draw a line perpendicular to MN passing through M. This line is the tangent to the curve at point M.

To construct the hyperbola with eccentricity e = 4/3 and distance from focus F to directrix DD = 75 mm, follow these steps:

1. Draw a vertical line as the directrix DD and a horizontal line as the axis of the hyperbola. Mark the intersection point as V'.
2. Locate the focus F on the axis such that the distance F-DD is 75 mm.
3. Divide the distance F-DD into 7 parts (e = 4/3, so 4+3=7). Mark the vertex V at 4 units from F and 3 units from DD.
4. Mark points 1, 2, 3, etc., on the axis beyond V. Draw vertical lines through these points.
5. With F as the center and radius equal to the distance from the point (e.g., 1) to the corresponding point on the axis, draw arcs to intersect the vertical line at 1' and 1''. Repeat this for all points to obtain the hyperbola curve.
6. To draw the tangent and normal at a point P located 35 mm from F, mark P on the curve.
7. Join P to F. Draw a line through F perpendicular to PF, intersecting the directrix at T.
8. The line joining T and P is the tangent to the hyperbola at P.
9. Draw a line perpendicular to the tangent TP passing through P; this is the normal to the curve at point P.