Conic Sections

Material Required: Modelling clay, chart paper, tracing paper, thin wire, divider, plane paper, pen, wooden board, nails string, filter paper, drawing pins, thread.

By Clay Modelling

By cutting through a cone at different angles to the base we can produce a family of interesting geometric curves are called conic sections. These curves are circle, parabola, hyperbola and ellipse.

Conic sections using modelling clay:

If a hollow cone is cut open by cutting along its slant edge and unrolled, we get a sector of a circle. Hence a cone can be rolled from a portion of circular paper. Cut a piece of chart paper into a circle using a divider. Cut away approximately a quarter of a circle by cutting along radial lines, retaining three quarters of it. This can be rolled into the shape of a cone, and the overlapping portion can be pasted or stapled together. Now make an identical cone in the same way from butter paper or tracing paper, but do not staple or paste it into the final shape. Place it inside the chart paper cone to form an inside layer. Now fill the cone with modelling clay. Press it down till it take a shape of a cone. Let the clay cone with the butter paper cover fall out of the paper cone. Peel off the butter paper. The cone is now ready for cutting. Stretch a thin wire holding the ends with both hands till it is taut. Use this wire to cut the cone at different angles to the base.

• A cut parallel to the base gives a circle at the section.
• A cut parallel to a sloping side gives a parabola.
• An ellipse is made by cutting through the cone at a slant.
• A cut parallel to the axis of the cone gives a hyperbola.

Ellipse by two pegs and Nails

Drawing an ellipse on board with two pegs or nails

A simple way to produce an ellipse is to use a thread and two nails. First hammer the two nails at certain distance. Now form a loop from the string about three times as long as the distance between the nails and place it across the nails. Now put the point of the pencil inside the loop and move it around the pins, keeping the pencil slanted so that the line is as smooth as possible. The curve that is obtained is an ellipse. Here the two nails form the foci of the ellipse. Ask the students why the sum of the distances from the two foci of every point on the ellipse is constant.

Ellipse, Parabola and Hyperbola by Paper Folding

One can obtain a beautiful ellipse by folding a circular piece of paper like the filter paper. Mark any point in the interior of the filter paper (preferably about 1 cm from the edge. Now fold the paper so that the circumference falls on F and press it down to crease it. Let the point on circumference which coincides with F be P1. Take the adjacent point on the circumference (say about 1 cm away from P1) and fold the paper again so that P1 falls on F. Crease the paper once again. Similarly fold the paper repeatedly so that a large number of points on the circumference fall on F. The whole circumference must be covered in this way. Examine the pattern of creases obtained A beautiful ellipse appears in the middle as an envelope of all the creases. The point F and the centre of the circle form the two foci of the ellipse.

A parabola can be obtain from a rectangular piece of paper in a similar manner. Mark a point F near one of the edges of the paper. Now fold repeatedly so that the edge close to F falls on F. The pattern of creases that emerge show a parabola. The point F forms the focus of the parabola and the edge which falls on F while folding forms the directrix of the parabola.

To fold a hyperbola take a piece of tracing paper (or some transparent paper which is not too thin and which creases well). Draw a circle on the paper and mark a point F outside the circle. Fold the paper repeatedly so that F falls on different points on the circumference of the circle. A pattern of two hyperbolas is obtained each of which is a reflection of the other. The center of the circle and the point F form the two foci of the hyperbola.

Exercise
In the paper folding activity for constructing conic sections, proof of the following can be given as exercise to the students:

1. The sum of the distances of any point on the ellipse from the two foci is constant. Hints: Each of the folded creases is a tangent to the ellipse. Study what happens with a single tangent. Folding creates a reflection with respect to the folded line of point F on the circumference. An important step is to identify the point on the tangent which touches the ellipse.
2. The sum of the distances of any point on the parabola from the focus and the directrix is constant.
3. The difference of the distances of any point on the hyperbola from the two foci is constant.

Parabola by Pins and Thread

We can make a hyperbola with thread and drawing pins. In this activity we push about six drawing pins into a piece of cardboard so that they lie at equal distances on a straight line. Make another identical line of drawing pins with the same gaps as the first line. The two lines must intersect to make an angle. Now with a piece of thread join the first pin of one line to the last (sixth) pin of the other to form a line segment between the pins. Join the second pin of the first line to the fifth pin of the second line. Continue joining the pins in this manner to obtain the hyperbola shown in the figure.