E.MATH- THIRD

• 2 millennia ago, Apollonius of Perga, the great greek geometer, studied the curves formed by the intersection of a plane and a double right curcular cone
• these curves were further known as CONIC SECTIONS, which are formed from the double right circular cone.
• The cone was thought as having 2 parts that extended infinitely in both directions.
• Generator of the cone

  • a line lying entirely on the cone
• All generators of a cone pass through the intersection of the 2 parts.
• VERTEX is the intersection of the 2 parts.
• Conic section is simpley called conic
• CONIC SECTION

  • According to Apollonius' study is a curve formed by the intersection of a plane an a double right circular cone.
• Types of conic section

  • Hyperbola

  • Parabola

  • Ellipse

  • Circle
• If the plane cuts one nape perpendicular to the cone axis, the figure formed is a circle.
• If the plane cuts one nape at an angle with the cone axis, the resulting figure is an ellipse.
• If the plane cuts one nape parallel to the side(generator) of a cone, the figure formed is a parabola.
• If the plane intersects both napes and is parallel to the cone axis, the figure is a hyperbola.
• Degenerated conics

  • Planes that pass through the vertex of the cone will intersect the cone in a point, a line or a pair of intersecting lines.
• If the cutting plane passes through the vertex perpendicular to the cone axis, we get a point. This is

  known as a degenerate ellipse.
• If the plane cuts through a side of the cones through the vertex, we have a single line. This is known as

  a degenerate parabola.
• If the cutting plane passes through the cone axis containing the vertex and two sides of the cone, we obtain two intersecting lines. This is known as a degenerate hyperbola.
• A conic is a set of points whose distances from a fixed point are in constant ratio to their distances from a fixed line that is not passing through the fixed point.
• CIRCLE

  • The most common conic found in nature. It is also one of the simplest mathematical curves.
• CIRCLE

  • It is the set of all points (x, y) in the plane whose distance from a fixed point is a constant. 
• The fix point is called the center of the circle and the constant distance from the center is called the radius of the circle.
• CENTER AT THE ORIGIN- STANDARD FROM

  X^2 + y^2 = r^2

  or standard form of equation of a circle whose center is at the origin

  with radius 𝑟.
• CENTER AT POING (H, K)

  • The equation of the circle whose center is at the point (ℎ, 𝑘) and with radius 𝑟

  (𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟2
• CENTER AT THE ORIGIN- GENERAL FORM

  • AX2 + AY2 + E =0
• CENTER AT POINT (H,K)- GEN FORMULA

  • X2 + Y2 +CX + DY+ E= 0
• To write general form to standard form, we need to use the steps in completing the square.

  i. Group the equation according to variables

  ii. Transpose the constant

  iii. Divide the whole equation by the value of a

  iv. Use the formula (𝑏

  2)2

  v. Add the answer of (𝑏

  2)2

  on both sides

  vi. Write the left side as the square of binomial