General Conic: Types, Equation, Formulas

General Conic: The locus of a moving point in a plane is a conic section if its distance from a stationary point (focus) is proportional to its perpendicular distance from a fixed line (i.e., directrix). Intersections of a plane and a double-napped right circular cone provide conic sections.

These curves are used in the design of telescopes, automotive headlights, and flashlight reflectors, among other things. The eccentricity of the conic is the constant ratio. A circle’s eccentricity is zero. It demonstrates how “circular” a curve is. The lower the curve, the greater the eccentricity. 

What is a General Conic?

A conic section (or simply conic) is a curve formed when the surface of a cone intersects a plane. The hyperbola, parabola, and ellipse are the three forms of conic sections. The circle, which is considered as the fourth type is a special case of an ellipse.

The intersection of a plane and a double right circular cone is a conic section. We can make several sorts of conics by adjusting the angle and location of the intersection. Circles, ellipses, hyperbolas, and parabolas are the four basic types. The intersections will not travel through the cone’s vertices.

A ‘conic‘ curve is one that is created by crossing a right circular cone with a plane. Euclidean geometry, it possesses unique features. The upper nappe and the lower nappe are separated by the cone’s vertex.

Different types of conic sections are obtained depending on the position of the plane that intersects the cone and the angle of intersection namely;

• Circle
• Ellipse
• Parabola
• Hyperbola

Types of General Conic Sections

Consider a fixed vertical line \(l\) intersecting at point \(V\) with another line \(m\) inclined at an angle \(\propto\):

We may create several sorts of conics by adjusting the angle and location of the intersection.

Circle is formed when a plane intersects right angle with the axis of the cone. Ellipse is formed when a plane intersects with the one of the side of cone and its axis is not perpendicular.

Parabola is formed when plane is parallel to one side of the cone and intersects one side of the double napped cone. Hyperbola is formed when a plane intersects two sides of the double napped cone.

Circle

If \(\beta=90^{\circ}\), the conic section formed is a circle as shown below.

Ellipse

If \(\alpha<\beta<90^{\circ}\), the conic section formed is an ellipse as shown in the figure below

Parabola

The conic section formed is a parabola as shown below, when \(\alpha=\beta\)

Hyperbola

The conic section is a hyperbola, if \(0 \leq \beta<\alpha\), then the plane intersects both nappes.

Parameters of General Conic

A conic is the locus of a moving point in a plane whose ratio of the distance from a stationary point to perpendicular distance from a fixed straight line is always constant.

• Focus: The focus of conic is the fixed point.
• Directrix: The directrix of the conic is a fixed straight line.
• Eccentricity: The eccentricity of the conic is the constant ratio. The letter \(e\) stands for it.

1. For circle, \(e=0\)
2. For parabola, \(e=1\)
3. For an ellipse, \(e<1\)
4. For hyperbola, \(e>1\)

Other important parameters are tabulated below:

General Equations of General Conic

The general equation of the conic is given by \(A x^{2}+B x y+C y^{2}+D x+E y+F=0\), where, \(A, B, C, D\), \(E\) and \(F\) are constants.

The shape of the related conic will change when the values of some of the variables change. To easily identify the type of conic represented by a particular equation, it is necessary to understand the differences in the equations.

Conic	Condition	Shapes
Circle	\(A\) and \(C\) are nonzero and equal, \(B=0\) and both have the same sign
Ellipse	\(A\) and \(C\) are nonzero and unequal and both have the same sign (\mathrm{B}^{2}-4 \mathrm{AC}>0\)
Parabola	\(A\) or \(C\) is zero \(\mathrm{B}^{2}-4 \mathrm{AC}=0\)
Hyperbola	\(A\) and \(B\) have different signs and are nonzero \(B^{2}-4 A C<0\)

Standard Forms and Formulas of General Conic

Applications of General Conic

Curves can be found in your car’s rear view mirror or in the enormous circular silver ones found at metro stations. Curves are employed in planetary motion studies, telescope design, satellite design, and reflector design, among other fields. Conics are curves that are generated when a plane intersects with a double-napped right circular cone.

Planets follow elliptical trajectories around the Sun, with one point of focus. Light beams are guided to the parabola’s focus using parabolic mirrors. Parabolic mirrors focus light beams for heating in solar ovens. Sound waves are concentrated by parabolic microphones. Look at the below pictures of glass at various situations, which shows the different general conics such as circle, parabola, ellipse, hyperbola etc.

Solved Examples – General Conic

Q.1. Check which of the geometrical figure has the following equation?
\(4 x^{2}-25 y^{2}-24 x+250 y-489=0\)

Ans: Given: Equation of conic is \(4 x^{2}-25 y^{2}-24 x+250 y-489=0\)

Comparing the above equation with the general equation of the conics \(A x^{2}+B x y+C y^{2}+\) \(\mathrm{D} x+\mathrm{E} y+\mathrm{F}=0\)

So, \(A=4\) and \(C=-25\)

\(A\) and \(C\) have different signs and nonzero. \((A, C \neq 0)\)

So, the given equation represents the equation of the ellipse.

Q.2. Determine the equation of the ellipse whose foci lie at \((\pm 5,0)\) and the eccentricity is \(\frac{1}{2}\).

Ans: Given: Foci of the ellipse at \((\pm 5,0)\)

So, the equation of the major axis of the ellipse is \(y=0\) or \(x-\) axis.

Thus, the equation of the ellipse is \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\), where \(a^{2}>b^{2}\)

We know that the formula for eccentricity of an ellipse, \(e=\frac{c}{a}\) and focus is \((\pm c, 0)\)

\(c=5\) and \(e=\frac{1}{2}\)

\(a=\frac{c}{e}=\frac{5}{\frac{1}{2}}=10\)

Now, \(c^{2}=\left(a^{2}-b^{2}\right)\)

\(\Rightarrow b^{2}=\left(a^{2}-c^{2}\right)=10^{2}-5^{2}=75\)

Hence, the equation of the ellipse is \(\frac{x^{2}}{100}+\frac{y^{2}}{75}=1\).

Q.3. Find the vertex, focus, eccentricity and length of axes and the latus rectum for a given hyperbola \(\frac{x^{2}}{9}-\frac{y^{2}}{16}=1\).

Ans: Given: Equation of the hyperbola is \(\frac{x^{2}}{9}-\frac{y^{2}}{16}=1\)

Comparing with the general equation of the hyperbola: \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\), we get \(a^{2}=9\) and \(b^{2}=16\)

So, \(a=3\) and \(b=4\)

\(c=\sqrt{a^{2}+b^{2}}=\sqrt{9+16}=\sqrt{2}=5\)

Q.4. The equation \(x^{2}+2 x y+y^{2}+4 x+3 y+5=0\) represents which conic?

Ans: Given: Equation is \(x^{2}+2 x y+y^{2}+4 x+3 y+5=0\)

Comparing the given equation with general equation of conic \(\mathrm{A} x^{2}+\mathrm{B} x y+\mathrm{C} y^{2}+\mathrm{D} x+\mathrm{E} y+\)

\(F=0\), we get

\(A=1, C=1\)

\(A\) and \(C\) are equal and have the same sign.

So, the given equation represents a circle.

Q.5. Find the focus and vertex coordinates, the directrix and axis equations, and the length of the latus rectum of the parabola \(y^{2}=16 x\).

Ans: Given: Equation is \(y^{2}=16 x\)

Comparing with the general equation \(y^{2}=4 ax\), we get

\(a=4\)

Summary

The conic sections are formed by the surface of a double right-circular cone intersecting with a plane. Parabola, circle, ellipse, and hyperbola are the four conic sections, while point, line, and two intersecting lines are the three degenerate forms.

\(\mathrm{A} x^{2}+\mathrm{B} x y+\mathrm{C} y^{2}+\mathrm{D} x+\mathrm{E} y+\mathrm{F}=0\) is the conventional form of a conic section equation, where A, B, C, D, E, F are real integers. There are different parameters such as focus, eccentricity, directrix, latus rectum etc. are associated with them.

We have different values and different graphs, formulas. A conic is the locus of a moving point in a plane whose ratio of the distance from a stationary point to perpendicular distance from a fixed straight line is always constant.

Frequently Asked Questions (FAQs)

Below are the frequently asked questions on General Conic:

Q.3. What is general conic form?
Ans: The standard form of the equation of a conic section is \(A x^{2}+B x y+C y^{2}+D x+E y+\) \(F=0\), where \(A, B, C, D, E, F\) are real numbers and \(A \neq 0, B \neq 0, C \neq 0\).

Q.4. What is the purpose of conics?
Ans: Conic sections can be seen in a variety of real-life scenarios. When we regard the Sun as one point of focus, the paths of the planets form ellipses around it. Parabolic mirrors aid in collecting light beams at the parabola’s focal point.

Q.5. Who discovered the general conic?
Ans: The conic section was discovered by Menaechmus, an ancient Greek mathematician. He also provided the first definition of a conic section.

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