 Hello and welcome to the session. In this session first we will discuss sections of a cone. Let this L be a fixed vertical line. Let this M be another line intersecting the fixed line L at a fixed point V and inclined to it at an angle alpha. Suppose that we rotate the line M around the line L in such a way that the angle alpha remains constant then the surface generated is a double-naped right circular hollow cone which is referred as cone and extending indefinitely far in both directions. So this is the cone. This point V is called the vertex. This line L is called the axis of the cone and the rotating line M is called the generator of the cone. This vertex separates the cone into two parts called the nape. This is the upper nape and this is the lower nape. If we take the intersection of a plane with the cone the sections are obtained is called the conic section. So we say basically conic sections are the curves obtained by intersecting a right circular cone by a plane. We obtain different kinds of conic sections depending on the position of the intersecting plane with respect to the cone and by the angle made by it with the vertical axis of the cone. Let this beta be the angle made by the intersecting plane with the vertical axis of the cone. Now the intersection of the plane with the cone can take place either at the vertex of the cone that is at this point or at any other part of the nape either below or above the vertex. Then we have when the plane cuts the nape other than the vertex of the cone. Then we have the falling situations like when the angle beta is 90 degrees then the section is a circle then when we have the angle beta is greater than alpha and less than 90 degrees then we say the section is an ellipse. Next situation is when we have angle alpha is equal to the angle beta then the section is a parabola. In three of these situations the plane cuts entirely across one nape of the cone. The next situation which can arises when the angle beta is greater than equal to zero and less than alpha then the plane cuts through both the nape and the curves of intersection is a hyperbola. Next we discuss degenerated conic sections when the plane cuts at the vertex of the cone. We have different cases like the following. The first case is when angle beta is greater than alpha and less than equal to 90 degrees then the section is a point. In the next case we have when angle beta is equal to angle alpha the plane contains a generator of the cone and the section is a straight line. This is the degenerated case of a parabola. Then we have one more case in which the angle beta is greater than equal to zero and less than alpha. The section is a pair of intersecting straight lines and this is a degenerated case of hyperbola. Now we discuss the conic section circle. Basically a circle is the set of all points in a plane that are equidistant from a fixed point in the plane. This fixed point O of the circle is called the center of the circle. Let's consider any point P on the circle then the distance from the center to any point on the circle is called the radius of the circle. So here we have OP is the radius of the circle. Let's consider the center has coordinates HK and a point P has coordinates XY and let OP be equal to R then we have equation of a circle with center at HK and radius R is given by X minus HD whole square plus Y minus K the whole square is equal to R square. Let's find the equation of the circle with center 3 minus 2 and radius equal to 5. Now from here we have H is equal to 3 and K is equal to minus 2 and the radius R is equal to 5. So the required equation of the circle is given by X minus H the whole square plus Y minus K the whole square is equal to R square that is X minus 3 the whole square plus Y plus 2 the whole square is equal to 25. So from here we get the equation of the circle is X square plus Y square minus 6X plus 4Y minus 12 is equal to 0. This is the final equation of the circle with center 3 minus 2 and radius 5. This completes the session hope you have understood what are the sections of a cone and the conic section circle.