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Crank
3 years ago
12

* Anyone please give me some information about the following;

Mathematics
2 answers:
Tema [17]3 years ago
7 0

Answer: See explanation below

Step-by-step explanation:

A conic is a curve obtained as the intersection of the surface of a cone with a plane. The 4 types of conic section are: 1. Hyperbola 2. The Parabola 3. The Ellipse and 4. Circle.

Hyperbola: When the plane cuts the cone at an angle closer to the axis than the side of the cone a hyperbola is formed.

Cones: A cone is a three-dimensional shape that goes in a diagonal directions from a flat base to a point called the apex or vertex. A cone is formed by a set of line segments connecting from the base to the apex.

Parabola: A parabola is the curve formed by the intersection of a plane and a cone, but only when the plane is at the same slope as the side of the cone.

Ellipse: An ellipse is formed by a plane intersecting a cone at an angle to its base. All ellipses have two focal points and a center and a major and minor axis.

Circles: A circle is formed when the plane is parallel to the base of the cone.

*Note: Some of these answers are PLAGIARISED from the internet as I looked these up and copied & pasted some information, so please paraphrase if turning this in.*

tatuchka [14]3 years ago
6 0

Answer:  see below

<u>Step-by-step explanation:</u>

Types of Conics are: Circles, Ellipses, & Hyperbolas.

Here is information about each one:

<u>Circle</u>: Two parabolas facing each other. Distance from the center to the vertices are exactly the same.

(x - h)² + (y - k)² = r²

  • (h, k) is the center of the circle
  • r is the radius of the circle
  • The 4 vertices are: (h, k+r), (h, k-r), (h + r, k), & (h - r, k)

<u>Ellipse:</u> Two parabolas facing each other. Distance from the center to the vertices are not the same.

(x - h)²/a² + (y - k)²/b² = 1

  • (h, k) is the center of the ellipse
  • a is the horizontal distance from the center to the vertices/co-vertices
  • b is the vertical distance from the center to the co-vertices/vertices
  • If a > b, then vertices are: (h+a, k), & (h-a, k) and co-vertices are (h, k+b), (h, k-b)
  • <em>If b > a, then the vertices and co-vertices are reversed </em>
  • Use |a² - b²| = c² to find the distance from the center to the foci.
  • If a > b, then Foci = (h+c, k) & (h-c, k). If b > a, then ± from the k-value

Example: (x - 1)²/9 + (y - 2)²/25 = 1

Center (h, k) = (1, 2)       a = √9 = 3       b = √25 = 5

Vertices (b > a): (1, 2+5) & (1, 2-5)  ==>  (1, 7) & (1, -3)

Co-vertices:      (1+3, 2) & (1-3, 2)  ==>  (4, 2) & (-2, 2)

Foci (b > a): c = √(25-9) = 4  --> (1, 2+4) & (1, 2-4)  ==>  (1, 6) & (1, -2)

<u>Hyperbola:</u> Two parabolas facing AWAY from each other. Distance from the center to the vertices may or may not be the same.

(x - h)²/a² - (y - k)²/b² = 1          or         (y - k)²/b² -  (x - h)²/a² = 1

               ↓                                                            ↓

opens left and right                                opens up and down

  • (h, k) is the center of the hyperbola
  • If x²-y², then vertices are: (h+a, k), & (h-a, k). <em>There are NO Co-vertices.</em>
  • If y²-x², then vertices are (h, k+b), (h, k-b). <em>There are NO Co-vertices.</em>
  • Use a² + b² = c² to find the distance from the center to the foci.
  • If x²-y², then Foci = (h+c, k) & (h-c, k). If y²-x², then ±c from the k-value
  • The slope of the asymptotes are ± b/a and pass through the center (h, k)

Example: (x - 1)²/9 - (y - 2)²/16 = 1

Center (h, k) = (1, 2)       a = √9 = 3       b = √16 = 4

Vertices (x²-y²): (1+3, 2) & (1-3, 2)  ==>  (4, 2) & (-2, 2)

Foci (x²-y²): c = √(9 + 16) = 5  --> (1+5, 2) & (1-5, 2)  ==>  (6, 2) & (-4, 2)

Asymptotes: m = ± 4/3    through (h, k) = (1, 2)     ==> y = ± 4/3(x - 1) + 2

               

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<h3>Two answers: 5, 7</h3>

====================================================

Explanation:

A drawing may be helpful to see what's going on. Check out the diagram below. This is one way of drawing out the two triangles. The locations of the points don't really matter, and neither does the the orientation of how you rotate things. What does matter is we have the right points connected to form the segments mentioned.

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For now, focus on triangle TIP only. In order to have this be isosceles, we must make TP = 5 or TP = 7.

If TP = 5, then it's the same length as TI.

If TP = 7, then it's the same length as PI.

In either case, we have exactly two sides the same length (the other side different) which is what it means for a triangle to be isosceles.

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Let's consider triangle TOP. For it to be isosceles, we must have two sides the same length. We already locked in TP to be either 5 or 7 in the previous section above. So there's no way that TP could be 11 units long to match up with PO = 11.

If TP = 5, then OT must also be 5 units long so that triangle TOP is isosceles.

If TP = 7, then OT = 7 for similar reasoning.

Either way, TP only has two choices on what it could be.

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In short, we basically just write the first two values given to us to get the two triangles to be isosceles. We can't use TP = 11 as it would make triangle TIP to be scalene (all sides are different lengths).

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y = \frac{1}{4}x - 5

Step-by-step explanation:

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