# data[].attributes.headline
فهم القطوع المخروطية

# data[].attributes.subject
فهم القطوع المخروطية

# data[].attributes.body
سيتمكن الطلاب من التعرف على أنواع القطوع المخروطية في وضعها الأساسي والتفريق بينها.

# data[].attributes.headline
السؤال الأساسي

# data[].attributes.subject
السؤال الأساسي

# data[].attributes.body
ما هي القطوع المخروطية وما الفرق بين أنواعها المختلفة؟

# data[].attributes.body
عرض تقديمي حول القطوع المخروطية باستخدام أمثلة مرئية لتوضيح الخصائص الأساسية.

# data[].attributes.attachment_link.title
What are Conic Sections? | Don't Memorise

# data[].attributes.body
قسّم الطلاب إلى مجموعات صغيرة لمناقشة واستنتاج الفروقات بين القطع المكافئ والناقص والزائد باستخدام ورقة عمل.

# data[].attributes.body
يعمل الطلاب على حل بعض الأسئلة ذات العلاقة لفهم وقوفاتهم على فهم القطوع.

# data[].attributes.headline
نشاط للطلاب المتقدمين

# data[].attributes.subject
نشاط للطلاب المتقدمين

# data[].attributes.body
تمرين تطبيقي يستخدم القطوع المخروطية في حل مسائل حقيقية مثل تصميم الأقمار الصناعية.

# data[].attributes.attachment_link.title
What your teachers (probably) never told you about the parabola, hyperbola, and ellipse

# data[].attributes.body
اطلب من الطلاب مناقشة رأيهم حول أهمية فهم القطوع المخروطية في التطبيقات اليومية.

# data[].attributes.headline
عرض أعمال الطلاب

# data[].attributes.subject
عرض أعمال الطلاب

# data[].attributes.body
يطلب من كل مجموعة عرض النتائج التي توصلت إليها من نشاط المجموعة.

# data[].attributes.body
اختبار مكون من 5 أسئلة لتقييم فهم الطلاب لأنواع القطوع المخروطية وخصائصها.

# data[].attributes.attachment_link.description
Conic Sections quiz for 9th grade students. Find other quizzes for Mathematics and more on Quizizz for free!

# data[].attributes.body
اطلب من الطلاب تلخيص أهم النقاط التي تم تعلمها خلال الدرس.

# data[].attributes.body
مهمة قصيرة للطلاب للإجابة عن السؤال: ما الذي استنتجته عن القطوع المخروطية اليوم؟

# data[].attributes.headline
ورقة عمل إضافية

# data[].attributes.subject
ورقة عمل إضافية

# data[].attributes.body
أكمل ورقة العمل المرفقة لتعزيز فهمك للقطوع المخروطية.

# data[].attributes.headline
قراءة إضافية

# data[].attributes.subject
قراءة إضافية

# data[].attributes.body
اقرأ المقال عن التطبيقات العملية للقطوع المخروطية واستعد لمناقشته في الحصة القادمة.

# data[].attributes.attachment_link.readable_text
27 October 2020

Read time: 8 minutes

Introduction

Conics or conic sections were studied by Greek mathematicians, with Apollonius of Pergo's work on their properties around 200 B.C.

Conics sections are planes, cut at varied angles from a cone. The shapes vary according to the angle at which it is cut from the cone.

As they are cut from cones, they are called Conies. Further, they have some common properties as they all belong to cones. These curved sections are related to.

Also read:
Conics in Real life

Here is a PDF that tells us more about conics in real life. They can think of these. Click on the download button to explore them.

What is Conic Section?

Conic section is a curve obtained by the intersection of the surface of a cone with a plane.

In Analytical Geometry, a conic is defined as a plane algebraic curve of degree 2. That is, it consists of a set of points which satisfy a quadratic equation in two variables. This quadratic equation may be written in matrix form. By this, some geometric properties can be studied as algebraic conditions.

Thus, by cutting and taking different slices(planes) at different angles to the edge of a cone, we can create a circle, an ellipse, a parabola, or a hyperbola, as given below

The circle is a type of ellipse, the other sections are non-circular. So, the circle is of fourth type.

Focus, Directrix and Eccentricity
The curve is also defined by using a point(focus) and a straight line (Directrix).

If we measure and let

a - the perpendicular distance from the focus to a point P on the curve,

and b - the distance from the directrix to the point P,

then a: b will always be constant.

\(a: b < 1\) for ellipse

\(a: b= 1\) for parabola as \(a= b\)

and \(a: b> 1\) for hyperbola.

Eccentricity: The above ratio a: b is the eccentricity.

Thus, any conic section has all the points on it such that the distance between the points to the focus is equal to the eccentricity times that of the directrix.

Thus, if eccentricity \(<1\), it is an ellipse.

if eccentricity \(=1\), it is a parabola.

and if eccentricity \(=1\), it is a hyperbola.

For a circle, eccentricity is zero. With higher eccentricity, the conic is less curved.

Latus Rectum

The line parallel to the directrix and passing through the focus is Latus Rectum.

Length of Latus Rectum = 4 times the focal length

Length \(=\frac{2b^2}{a}\) where \(a =\frac{1}{2}\) the major diameter

and \(b =\frac{1}{2}\) the minor diameter.

= the diameter of the circle.

Ellipse has a focus and directrix on each side i.e., a pair of them.

Equations

General equation for all conics is with cartesian coordinates x and y and has \(x^2\) and \(y^2\) as

the section is curved. Further, x, y, x y and factors for these and a constant is involved.

Thus, the general equation for a conic is

\[Ax^2 + B x y + C y^2+ D x + E y + F = 0\]

Using this equation, following equations are obtained:

For ellipse, \(x^2a^2+y^2b^2=1\)

For hyperbola, \(x^2a^2-y^2b^2=1\)

For circle, \(x^2a^2+y^2a^2=1\) (as radius is a)

Parabola in Real Life

Parabola is obtained by slicing a cone parallel to the edge of the cone. It is of U - shape as a stretched geometric plane. This formula is \(y =x^2\) on the x - y axis.

Mathematician Menaechmus derived this formula.

Parabola is found in nature and in works of man.

A ball thrown high, follows a parabolic path.

A roller coaster takes the path of rise and fall of a parabolic track of the sea.

Ellipses in Real Life

Paul's Cathedral is an elliptical shaped structure to facilitate talking at one end is heard at the other end using the property of ellipse.

Whispering galleries at US Statutory capital and St. Paul's Cathedral, London demonstrates the property of the ellipse that one's whisper from one focus can be heard at the other focus by only a person to whom it is sent

>

Hyperbolas in Real Life

Satellite systems, Radio systems use hyperbolic functions.

Lens, monitors, and optical glasses are of hyperbola shape.

Summary
Conic or conical shapes are planes cut through a cone. Based on the angle of intersection, different conics are obtained. Parabola, Ellipse, and Hyperbola are conics. Circle is a special conic. Conical shapes are two dimensional, shown on the x, y axis. Conic shapes are widely seen in nature and in man-made works and structures. They are beneficially used in electronics, architecture, food and bakery and automobile and medical fields.
Source: google content

Frequently Asked Questions (FAQs)

What are the 4 types of conic sections?

According to the angle of intersection between a plane and a cone, four different conic sections are obtained. They are Parabola, Ellipse, Hyperbola, and Circle. They are two dimensional on the x-y axis.

How do we obtain Conic Sections?

Conic section involves a cutting plane, surface of a double cone in hourglass form and the intersection of the cone by the plane. According to the angle of cutting, that is, light angle, parallel to the edge and deep angle, ellipse, parabola and hyperbola respectively are obtained. Circle is also conic, and it is cut parallel to the circular bottom face of the cone.

How do you find the area of an ellipse?

Area of an ellipse is \((a \times b \times π)\) sq. units.
Source:i.pinimg.com where a = length of major axis of ellipse b = length of minor axis of ellipse.

What are some real-life applications of conics?

Planets travel around the Sun in elliptical routes at one focus. Mirrors used to direct light beams at the focus of the parabola are parabolic. Parabolic mirrors in solar ovens focus light beams for heating. Sound waves are focused by parabolic microphones. Car headlights and spotlights are designed based on parabola's principles. The path travelled by objects thrown into air is parabolic. Hyperbolas are used in long range navigation systems called LORAN. Telescopes use parabolic mirrors.

# data[].attributes.attachment_link.title
Applications of Conics in Real Life | Conic Sections

# data[].attributes.attachment_link.description
Learn about the different uses and applications of Conics in real life. Parabolas in real life, Ellipses in real life, Hyperbolas in real life.

# data[].attributes.headline
استنتاج خصائص القطوع المخروطية

# data[].attributes.subject
استنتاج خصائص القطوع المخروطية

# data[].attributes.body
سيتمكن الطلاب من تحديد الخصائص الهندسية لكُلٍ من القطع الناقص، القطع المكافئ، والقطع الزائد واستخدامها في التطبيقات الرياضية.