Conic Sections

  • Introduction to Conic Section

    A conic section is defined as the curve of intersection of a plane with a right circular cone of two nappes. There are three types of curves that occur in this way: the parabola, the ellipse, and the hyperbola. The resulting curves depend upon the inclination of the axis of the cone to the cutting […]

  • Applications of Conic Sections

    There are many applications of conic sections to both pure and applied mathematics. We shall mention a few of them. The orbits of planets and satellites are ellipses. Ellipses are used in making machine gears. Arches of bridges are sometimes elliptical or parabolic in shape. The path of a projectile is a parabola if motion […]

  • Definition of Conic

    A conic is the set of all points in a plane such that the distance of from fixed point is in a constant ratio to the distance of from a fixed line which does not contain the fixed point. The fixed point is called a focus of a conic, and the fixed line is called […]

  • Definition of Circle

    To understand the circle considers the two dimensions -Plane. The set of all points in a plane which are equidistant from some fixed point in the plane is called a circle. The fixed point is called the centre of the circle. The fixed distance from the centre to the points of the circle is called […]

  • Standard Equation of a Circle

    Let be any point of the circle as shown in the diagram, then by the definition of circle, the distance of point from must be equal to the radius of the circle . i.e. . As we know that distance formula from the analytic geometry as Now we shall use this formula to establish the […]

  • Parametric Equations of Circle

    Draw a circle with centre at and radius is equal to which is the fixed distance from the centre of the circle. Now let be any point of the circle as shown in the diagram. Draw a perpendicular from point on X-axis meet at the point . Consider the triangle which is a right angled […]

  • General Form of Equation of Circle

    Consider the equation of circle in general form is Where are any constant values. Rearrange the terms of the above equation (i) of circle, we have In this equation we use the method of completing squares, so for this we need to add and on both sides of the equation (ii). i.e. Compare this equation […]

  • Equation of Circle with Endpoints of Diameter

    Let and be the end points of the diameter of the circle as shown in the diagram. Let be any point of the circle. Connecting the points and with the point and makes an angle between them. First we find the slopes of the lines and as: Slope of the line Slope of the line […]

  • Equation of a Circle through Three Points

    Consider the general equation circle is given by If the given circle is passing through three non-collinear points, say, , and , then these points must satisfy the general equation of circle. Now put the above three points in the given equation of circle, i.e. To evaluate the equation of required circle, we must the […]

  • Equation of a Circle Given Two Points and Tangent Line

    Consider the general equation circle is given by If the given circle is passing through two points, say, and , then these points must satisfy the general equation of circle. Now put these two points in the given equation of circle, i.e. Also the given straight line touches the circle at one point as shown […]

  • Two Circles Touch Internally

    If two given circles are touches internally with each other, take an example to understand the concept of internally touches circles. Consider the given circles and Let and be the centre and radius of the circle (i) respectively, now to find centre and radius compare the equation of circle with general equation of circle to […]