Chapter 6 Maths Class 12 Notes

• Important chapter 6 maths class 12 important questions for Intermediate part-II students.
• Introduction: Conic sections, or simply conics, are the curves obtained by cutting a (double) right circular cone by a plane. Let RS be a line through the centre C of a given circle and perpendicular to its plane. Let A be a fixed point on RS. All lines through A and points on the circle generate a right circular cone. The lines are called rulings or generators of the cone. The surface generated consists of two parts, called nappes, meeting at the fixed point A, called the vertex or apex of the cone. The line RS is called axis of the cone.
• If the cone is cut by a plane perpendicular to the axis of the cone, then the section is a circle.
• The size of the circle depends on how near the plane is to the vertex of the cone. If the plane passes through the vertex A, the intersection is just a single point or a point circle. If the cutting plane is slightly tilted and cuts only one nappe of the cone, the resulting section is an ellipse. If the intersecting plane is parallel to a generator of the cone, but intersects its one nappe only, the curve of intersection is a parabola. If the cutting plane is parallel to the axis of the cone and intersects both of its nappes, then the curve of intersection is a hyperbola.
• Important chapter 6 maths class 12 solutions pdf for Intermediate part-II students.
• The Greek mathematicians Apollonius' (260-200 B.C.) and Pappus (early fourth century) discovered many intersecting properties of the conic sections. They used the methods of Euclidean geometry to study conics.
• Tangents and Normals: The normal to a curve at a point on the curve is perpendicular to the tangent through the point of tangency.
• Remarks: An equation of the tangent at the point (x1,y1) of any conic can be written by making replacements in the equation of the conic.
• Important class 12 maths chapter 6 solutions pdf for Intermediate part-II students.
• Theorem: To show that a straight line cuts a conic, in general, in two points and to find the condition that line be a tangent to the conic let y = mx + c cuts the conics (i) y^2 = 4ax
• Translation of Axis:  If a point P has coordinates (x,y) referred to xy- system and has coordinated (x,y) referred to the translated axes 0'x,0'y through 0'(h,k).
• Important chapter 6 maths class 12 exercise 6.4 for Intermediate part-II students.
• Rotation of Axes: Let xy-coordinate system be given. We rotate Ox, Oy about the origin through an angle (0<0<90°) so that the new axes are Ox and Oy as shown in the fig. Let a point P(x,y) referred coordinates, P(x,y) referred to XY- coordinate system. We have to find XY-coordinates in terms of the given coordinates (x,y). Let a be the measure of inclination of OP. From P, draw PM Lar to Ox and PQ Lar to Ox. Let l op l=r
• Important chapter 6 maths class 12 exercise 6.5 for Intermediate part-II students.
• The general equation of Second degree:  Ax^2 + By^2 + Gx+Fy+C= 0 The most general eq. of the second degree ax²+2hxy+by+2gx + 2 fy+c=0 represents a comic Here is called the → (1) discriminant (1) represents
• (i) An ellipse or circle if h^2-ab< 0
• (ii) A parabola if h^2-ab=0
• (iii) A hyperbola if h^2-ab
• Important chapter 6 maths class 12 exercise 6.6 for Intermediate part-II students.
• Equations of transformation are X = XCOSO-Y sin (1) Y = X Sin O+YCOSO (ⅱ) Solving (i) & (ii) for X,Y we find X=xcosO+ y sin & 8 Y = -x Sino + y coso