Conic Sections mathematics project term 2 Grade 12

1. Conic Sections mathematics project term 2 Grade 12

2. Task 1 : Why are they called conic sections? Each of the geometric figures are obtained by intersecting a double-napped right circular cone with a plane. Thus, the figures are called conic sections or conics. Definition:A conic section is the intersection of a plane and a cone. By changing the angle and location of intersection, we can produce a circle, ellipse, parabola or hyperbola Which are generally the different ways you can divide plane and a cone in a symmetrical way.

3. Task 1 : Circle •  If the plane cuts completely across one nappeof the cone and is perpendicular to the axis of the cone, the curve of the section is called a circle. AND It’s a special case of Ellipse which you will see the next slide.

4. Task 1 : Ellipse • If the plane isn't perpendicular to the axis of the cone, it is called an ellipse. • An ellipse is the set of all points in a plane, the sum of the distances from two fixed points in the plane is constant. • Many comets have elliptical orbits.

5. Task 1 : Parabola • If the plane doesn't cut across one entire nappe or intersect both nappes, the curve of the intersection is called a parabola. • A parabola is the set of all points in a plane equidistant from a fixed point and a fixed line in the plane.

6. Task 1 : Hyperbola • If the plane cuts through both nappes of the cone, the curve is called a hyperbola. • The hyperbola is the set of all points in a plane. The difference of whose distance from two fixed points in the plane is the positive constant.

7. Task 1 : The use of conic sections • The Greeks discovered the properties that define conics in terms of points and lines. These properties are important tool for current day problems with the behaviors of atoms, molecules and outer space. Particles that move under the influence of an inverse square force field has a path that is described by conic sections. 

8. Task 3 Parabola • You can find the equation of a line by knowing two points from that line, know to find and equation of parabola you need to know three points. Find the equation of a parabola that pass through (0,3), (-2, 7) and (1, 4). 5 . 4a-2(1-a)=4 4a-2+2a=4 6a=6 a=1 b=0 1. (0,3)  y = ax2 + bx + c 3 = a(0)2 + b(0) + c c=3 2.(-2, 7)  y = ax2 + bx + c 7 = a(-2)2 + b(-2) + 3 4a-2b=4 3. (1, 4) y = ax2 + bx + c 4 = a(1)2 + b(1) + 3 a+b=1 4. 4a-2b=4 a+b=1 b=1-a Equation :  y =(1)x +(0)x + c y = x2+ 3

9. Task 3 Circle • If you have a line equation 𝑥𝑥𝑦𝑦and circle equation 𝑥𝑥𝑦𝑦. How many points the graphs of these two equations have in common. x=-2y+2 (-2y+2)2 + y2 = 25 5y2-8y-21 = 0 y1=3 x1=-4(-4, 3) y2=-1.4x2=4.8(4.8, -1.4)

10. Task 3 • Now Graphically explore the all cases of line and circle intersections in the plane. A=(-4,3) B=(4.8, -1.4)

11. Task 4 Physics: • The path of any thrown ball is parabola. Suppose a ball is thrown from ground level, reach a maximum height of 20 meters of and hits the ground 80 meters from where it was thrown. Find the equation of the parabolic path of the ball, assume the focus is on the ground level. Vertex (0,20) c= |1/a|=|1/80|=80 Y=a(x-h)2+k Y=-1/80(x-0)2+20(because it’s vertical) Y=-1/80x2+20

12. Task 4 Halley’s Comet • It takes about 76 years to orbit the Sun, and since it’s path is an ellipse so we can say that its movements is periodic. But many other comets travel in paths that resemble hyperbolas and we see it only once. Now if a comet follows a path that is one branch of a hyperbola. Suppose the comet is 30 million miles farther from the Sun than from the Earth. Determine the equation of the hyperbola centered at the origin for the path of the comet. Final equation: C=38 (76/2) a=15 (30/2) C2=a2+b2 382=152+b2 1444=225+b2 b2=1444-225 b2=1219

13. Done by: • EimanAlneyadi • Alanunodyousef • Mouza Al ahbabi • AlaounodThani 12.53