Mathematics project for class 12 on graphs

class 11 project for maths

1. The graph of sin The graph for a sin function is called a sinusoidal wave. The graph of sin x/2 and 2 sin x are half and twice in magnitude as the graph of sin x respectively . The graph of sin2x can be plotted using various values of x. For example: For x = 30°, sin2x = sin60° = √3/2 , 2 sin x =2 * (sin 30) For x = 45°, sin2x = sin90° = 1.

2. hyperbola A hyperbola, in analytic geometry, is a conic section that is formed when a plane intersects a double right circular cone at an angle such that both halves of the cone are intersected. This intersection of the plane and cone produces two separate unbounded curves that are mirror images of each other called a hyperbola.

3. A hyperbola, in analytic geometry, is a conic section that is formed when a plane intersects a double right circular cone at an angle such that both halves of the cone are intersected. This intersection of the plane and cone produces two separate unbounded curves that are mirror images of each other called a hyperbola.

4. Parts of a Hyperbola Let us check through a few important terms relating to the different parameters of a hyperbola. Foci of hyperbola: The hyperbola has two foci and their coordinates are F(c, o), and F'(-c, 0). Center of Hyperbola: The midpoint of the line joining the two foci is called the center of the hyperbola. Major Axis: The length of the major axis of the hyperbola is 2a units.] Minor Axis: The length of the minor axis of the hyperbola is 2b units. Vertices: The points where the hyperbola intersects the axis are called the vertices. The vertices of the hyperbola are (a, 0), (-a, 0). Latus Rectum of Hyperbola: The latus rectum is a line drawn perpendicular to the transverse axis of the hyperbola and is passing through the foci of the hyperbola. The length of the latus rectum of the hyperbola is 2b2/a.

5. Transverse Axis: The line passing through the two foci and the center of the hyperbola is called the transverse axis of the hyperbola. Conjugate Axis: The line passing through the center of the hyperbola and perpendicular to the transverse axis is called the conjugate axis of the hyperbola. Eccentricity of Hyperbola: (e > 1) The eccentricity is the ratio of the distance of the focus from the center of the hyperbola, and the distance of the vertex from the center of the hyperbola. The distance of the focus is 'c' units, and the distance of the vertex is 'a' units, and hence the eccentricity is e = c/a.

6. Focal property The focal property of a hyperbola is the characteristic property. Based on this property, one can define a hyperbola as a curve on a plane such that the modulus of the difference of distances from any point of the curve to the two fixed points (foci) on a plane is the constant value. Other properties of hyperbolas include:- There are two foci for the hyperbola.- The foci lie on the axis of the hyperbola.- The foci of the hyperbola is equidistant from the center of the hyperbola.- The foci of hyperbola and the vertex of hyperbola are collinear.

7. construction To construct a hyperbola using its focal property, we can follow these steps: 1. Draw two perpendicular lines intersecting at point O. 2. Mark two points F1 and F2 on these lines such that OF1 = OF2 = c (where c is a constant). 3. Draw a line passing through F1 and F2. 4. Take any point P on this line and mark its distance from F1 as d1 and its distance from F2 as d2. 5. Draw perpendiculars from P to both lines containing F1 and F2. 6. Let these perpendiculars intersect these lines at points A and B respectively. 7. Join AB. This line is called the transverse axis of the hyperbola. 8. Draw perpendiculars to AB at points A and B. 9. Let these perpendiculars intersect the line containing F1 and F2 at points C and D respectively. 10. Join CD. This line is called the conjugate axis of the hyperbola.