Conic Section (Ellipses and Parabolas) Notes | Junior Cycle Graphics

Conic sections are curves formed when a flat cutting plane slices through a cone at different angles. In the Junior Cycle examination, you must identify cone cuts, locate missing axes and focal points, and construct accurate ellipses and parabolas.

Conic Sections and Cone Cuts

A right circular cone has a flat circular base, an apex, and a curved surface made of straight lines called generators running from apex to base. Slicing the cone with a cutting plane produces different cross-sections depending on the angle of the cut.

Cutting plane position	Section produced
Parallel to the circular base	Circle
Inclined to the base and cutting every generator, without passing through the apex	Ellipse
Parallel to an extreme generator (slant edge)	Parabola
Passing through the apex and cutting across the base	Triangle

Four cones show cuts producing a circle, ellipse, parabola and triangle.

Parts of the Ellipse

An ellipse is a closed, symmetrical curve defined by two perpendicular axes meeting at the centre.

Major axis: the longest diameter across the ellipse, passing through both focal points. Half this length is the semi-major axis.
Minor axis: the shortest diameter through the centre, perpendicular to the major axis. Half this length is the semi-minor axis.
Focal points: two fixed points, $F_1$ and $F_2$ , lying on the major axis.
Focal radii: the two lines $PF_1$ and $PF_2$ joining any point $P$ on the curve to the foci. Their sum always equals the major axis ( $|PF_1| + |PF_2| = \text{major axis}$ ).
Minor-axis property: the distance from either end of the minor axis to either focal point equals half the major axis.

Constructing an Ellipse and Finding Axes

To find an unknown semi-minor axis, swing an arc of radius equal to half the major axis from a focal point to cut the minor axis line.

An arc centred at F₁ with radius equal to half the major axis meets the perpendicular minor axis at B.

The concentric circles method constructs an ellipse using two auxiliary circles with radii equal to the semi-major and semi-minor axes:

Divide both circles into 12 sectors using a $30^\circ/60^\circ$ set square. Projecting parallel to the minor axis from the major circle and parallel to the major axis from the minor circle locates points on the curve.

Matching points on two concentric circles project vertically and horizontally to meet at a point on the ellipse.

To construct a tangent at point $P$ on the curve, join $P$ to $F_1$ and $F_2$ . Bisect angle $\angle F_1PF_2$ to find the normal, then draw the tangent through $P$ at $90^\circ$ to that normal.

At P on an ellipse, the normal bisects the angle between the focal radii, and the tangent is perpendicular to the normal.

The Parabola and the Rectangle Method

A parabola is an open, symmetrical curve. Every point on it maintains equal distance from a fixed point (the focus) and a fixed line (the directrix). Its vertex is the turning point where the curve meets its axis of symmetry.

A point P on a parabola is equally distant from focus F and the perpendicular foot D on the directrix.

The rectangle method constructs a parabola within a rectangle centred on its axis of symmetry:

Divide the outer vertical side and half-base into the same number of equal parts. Radiate lines from vertex $V$ to the side points, and project lines parallel to the axis from the base points. Their matching intersections locate points on the curve.

Rays from the top-centre vertex intersect vertical projections from matching half-base divisions to locate a parabolic arch.

Key terms

Generator: A straight line on the curved surface of a cone running from the apex to the edge of the base.
Major Axis: The longest diameter passing through the centre of an ellipse and containing both focal points.
Minor Axis: The shortest diameter passing through the centre of an ellipse, perpendicular to the major axis.
Focal Radii: The lines joining any point on an ellipse to both focal points; their combined length always equals the major axis.
Auxiliary Circles: Two concentric circles drawn from the centre of an ellipse with radii equal to the semi-major and semi-minor axes.
Tangent: A straight line that touches a curve at one point only without crossing it.
Normal: A straight line drawn perpendicular (90°) to the tangent at the point of contact.
Vertex: The point where a conic curve meets its axis of symmetry, such as the turning point of a parabola.

Check yourself

Which conic section is formed by a cutting plane parallel to an extreme generator of a cone?
A parabola.
What is the sum of the focal radii |PF1| + |PF2| for any point P on an ellipse?
It is always equal to the full length of the major axis.
How can you find the position of the focal points F1 and F2 when both axes are drawn?
Set a compass to half the major axis, place the point at an end of the minor axis, and swing arcs cutting the major axis.
What angle does the normal line make with the tangent at the point of contact?
Exactly 90° (they are perpendicular).

Conic Section (Ellipses and Parabolas)