Problem 1 : Draw ellipse by concentric circle method. Take major axis 100 mm and minor axis 70 mm long.
Construction
Step 1. Draw both the major & minor axes as perpendicular bisectors of each other.
Step 2. Taking their intersecting point as a center, draw two concentric circles considering both as respective diameters.
Step 3. Divide both circles in 12 equal parts & name as shown.
Step 4. From all points of outer circle draw vertical lines downwards and upwards respectively. From all points of inner circle draw horizontal lines to intersect those vertical lines.
Step 5. Join all these intersecting lines along with the ends of both axes in smooth possible curve. It is required ellipse.
Step 1. Draw both the major & minor axes as perpendicular bisectors of each other.
Step 2. Taking their intersecting point as a center, draw two concentric circles considering both as respective diameters.
Step 3. Divide both circles in 12 equal parts & name as shown.
Step 4. From all points of outer circle draw vertical lines downwards and upwards respectively. From all points of inner circle draw horizontal lines to intersect those vertical lines.
Step 5. Join all these intersecting lines along with the ends of both axes in smooth possible curve. It is required ellipse.
Problem 2 : Draw ellipse by Rectangle method. Take major axis 100 mm and minor axis 70 mm long .
Construction
Step 1. Draw a rectangle taking major and minor axes as sides. In this rectangle draw both axes as perpendicular bisectors of each other.
Step 2. For construction, select upper left part of rectangle. Divide vertical small side and horizontal long side into same number of equal parts.( here divided in four parts)
Step 3. Now join all vertical points 1,2,3,4, to the upper end of minor axis. And all horizontal points i.e.1,2,3,4 to the lower end of minor axis.
Step 4. Then extend D-1 line upto C-1 and mark that point. Similarly extend D-2, D-3, D-4 lines up to C-2, C-3, & C-4 lines.
Step 5. Mark all these points properly and join all along with ends A and C in smooth possible curve. Do similar construction in right side part along with lower half of the rectangle. Join all points in smooth curve. It is required ellipse.
Step 1. Draw a rectangle taking major and minor axes as sides. In this rectangle draw both axes as perpendicular bisectors of each other.
Step 2. For construction, select upper left part of rectangle. Divide vertical small side and horizontal long side into same number of equal parts.( here divided in four parts)
Step 3. Now join all vertical points 1,2,3,4, to the upper end of minor axis. And all horizontal points i.e.1,2,3,4 to the lower end of minor axis.
Step 4. Then extend D-1 line upto C-1 and mark that point. Similarly extend D-2, D-3, D-4 lines up to C-2, C-3, & C-4 lines.
Step 5. Mark all these points properly and join all along with ends A and C in smooth possible curve. Do similar construction in right side part along with lower half of the rectangle. Join all points in smooth curve. It is required ellipse.
PROBLEM 1 : A BALL THROWN IN AIR ATTAINS 100 M HIEGHT AND COVERS HORIZONTAL DISTANCE 150 M ON GROUND. Draw the path of the ball (projectile)
Construction
Step 1. Draw rectangle of above size and divide it in two equal vertical parts
Step 2. Consider left part for construction. Divide height and length in equal number of parts and name those 1,2,3,4,5& 6
Step 3. Join vertical 1,2,3,4,5 & 6 to the top center of rectangle
Step 4. Similarly draw upward vertical lines from horizontal1,2,3,4,5 And wherever these lines intersect previously drawn inclined lines in sequence Mark those points and further join in smooth possible curve.
Step 5. Repeat the construction on right side rectangle also.Join all in sequence. This locus is Parabola .
Step 1. Draw rectangle of above size and divide it in two equal vertical parts
Step 2. Consider left part for construction. Divide height and length in equal number of parts and name those 1,2,3,4,5& 6
Step 3. Join vertical 1,2,3,4,5 & 6 to the top center of rectangle
Step 4. Similarly draw upward vertical lines from horizontal1,2,3,4,5 And wherever these lines intersect previously drawn inclined lines in sequence Mark those points and further join in smooth possible curve.
Step 5. Repeat the construction on right side rectangle also.Join all in sequence. This locus is Parabola .
Problem 2 : Draw an isosceles triangle of 100 mm long base and 110 mm long altitude. Inscribe a parabola in it by method of tangents.
Construction
Step 1 . Construct triangle as per the given dimensions.
Step 2. Divide it’s both sides in to same no.of equal parts.
Step 3. Name the parts in ascending and descending manner, as shown.
Step 4. Join 1-1, 2-2,3-3 and so on.
Step 5. Draw the curve as shown i.e. tangent to all these lines. The above all lines being tangents to the curve, it is called method of tangents .
Step 1 . Construct triangle as per the given dimensions.
Step 2. Divide it’s both sides in to same no.of equal parts.
Step 3. Name the parts in ascending and descending manner, as shown.
Step 4. Join 1-1, 2-2,3-3 and so on.
Step 5. Draw the curve as shown i.e. tangent to all these lines. The above all lines being tangents to the curve, it is called method of tangents .
PROBLEM 1 : Point F is 50 mm from a line AB. A point P is moving in a plane such that the ratio of it's distances from AB and line F remains constant and equal to 2/3. Draw locus of point P. { Eccentricity = 2/3 }
Construction
Step 1 . Draw a vertical line AB and point F 50 mm from it. Divide 50 mm distance in 5 parts.
Step 2. Name 2nd part from AB as V. It is 20mm and 30mm from AB and F line resp. It is first point giving ratio of it’s distances from AB and F 2/3 i.e 20/30.
Step 3. Form more points giving same ratio such as 30/45, 40/60, 50/75 etc. Taking 30,40 and 50 mm distances from line AB, draw three vertical lines to the right side of it.
Step 4. Now with 45, 60 and 75 mm distances in compass cut these lines above and below, with F as center.
Step 5. Join these points through V in smooth curve. This is required locus of P.It is the required hyperbola.
Step 1 . Draw a vertical line AB and point F 50 mm from it. Divide 50 mm distance in 5 parts.
Step 2. Name 2nd part from AB as V. It is 20mm and 30mm from AB and F line resp. It is first point giving ratio of it’s distances from AB and F 2/3 i.e 20/30.
Step 3. Form more points giving same ratio such as 30/45, 40/60, 50/75 etc. Taking 30,40 and 50 mm distances from line AB, draw three vertical lines to the right side of it.
Step 4. Now with 45, 60 and 75 mm distances in compass cut these lines above and below, with F as center.
Step 5. Join these points through V in smooth curve. This is required locus of P.It is the required hyperbola.
Problem 2 : A sample of gas is expanded in a cylinder from 10 unit pressure to 1 unit pressure.Expansion follows law PV=Constant.If initial volume being 1 unit, draw the curve of expansion. Also Name the curve.
Construction
Step 1. Take pressure on vertical axis and Volume on horizontal axis.
Step 2. Divide both the axes in ten equal parts. Name these parts from 1 to 10.
Step 3. Now according to the table given below locate the points on the graph.
Step 1. Take pressure on vertical axis and Volume on horizontal axis.
Step 2. Divide both the axes in ten equal parts. Name these parts from 1 to 10.
Step 3. Now according to the table given below locate the points on the graph.