Conic Sections

A conic section is defined as the curve of the intersection of a plane with a right circular cone of... Click here to read more

There are many applications of conic sections in both pure and applied mathematics. Here we shall discuss a few of them.... Click here to read more

A conic is the set of all points in a plane such that the distance of from a fixed point... Click here to read more

To understand the circle, consider the two dimensions -Plane. The set of all points in a plane which are equidistant... Click here to read more

Let be any point of the circle as shown in the diagram, then by the definition of a circle, the... Click here to read more

Draw a circle with centre at and with a radius equal to which is the fixed distance from the centre... Click here to read more

Consider the equation of a circle in general form is Where are any constant values. If we rearrange the terms... Click here to read more

Let and be the end points of the diameter of the circle as shown in the diagram. Let be any... Click here to read more

Consider the general equation a circle is given by If the given circle is passing through three non-collinear points, say,... Click here to read more

Consider the general equation a circle is given by If the given circle is passing through two points, say and... Click here to read more

Consider the general equation a circle is given by If the given circle is passing through two points, say and... Click here to read more

If two given circles are touching each other internally, use this example to understand the concept of internally toucheing circles.... Click here to read more

To understand the concept of two given circles that are touching  each other externally, look at this example. Consider the given circles... Click here to read more

In this tutorial we find the equation of circles with both axes touching, i.e. the X-axis and Y-axis, with any given... Click here to read more

In plane geometry, a straight line is called a tangent to a circle. If this line meets the circle at... Click here to read more

Consider the equation of circle with centre at origin and radius . Then the equation of this circle is written... Click here to read more

Consider the equation of a circle with center at origin and radius . Then the equation of this circle is written... Click here to read more

Consider the equation of a circle with center at origin and radius . Then the equation of this circle is... Click here to read more

To find the points where the given line cuts the circle, we take an example as follows: Example: Find the... Click here to read more

The equations of tangent and normal to the circle at the point are defined by and respectively. Equation of Tangent... Click here to read more

The point lies outside, on or inside the circle according to , and respectively. The given equation of circle is The... Click here to read more

Two tangents can be drawn to a circle from any point . The tangents are real and distinct, coincident or... Click here to read more

We will find the equation of tangent lines drawn from the point to the circle given by Now to solve this... Click here to read more

Let the tangent drawn from the point meet the circle at the point as shown in the given diagram. The... Click here to read more

The length of the diameter of the circle is equal to . Consider the equation of a circle is given... Click here to read more

The perpendicular dropped from the center of a circle on a chord bisects the chord. Consider the equation of the... Click here to read more

The normal lines of a circle passes through the center of the circle. Consider the equation of the circle Since... Click here to read more

Two congruent chords of a circle are equidistant from its center. NOTE: Two chords are said to be congruent if... Click here to read more

Consider the equation of the circle with the center at the origin is given by Let be any diameter of the... Click here to read more

The perpendicular dropped from a point of a circle on a diameter is a mean proportional between the segments into... Click here to read more

The midpoint of the hypotenuse of a right triangle is the circumcenter of the triangle. Consider the equation of the... Click here to read more

In this tutorial we shall study the parabola, which is obtained by the intersection of the plane and a right... Click here to read more

We shall derive the equation of a parabola from the definition. In order for this equation to be as simple as... Click here to read more

We see that for the equation the parabola opens to the right if and to the left if . The... Click here to read more

The point of parabola closed to focus is the vertex. Let the given parabola be The vertex and the focus... Click here to read more

Example 1: Find an equation of the parabola having its focus at and as its directrix on the line .... Click here to read more

In plane geometry, a straight line is called the tangent to the parabola. If this line meets the parabola at... Click here to read more

The line intersects the parabola at two points maximum and the condition for such intersection is that . Consider the... Click here to read more

The line does not intersect the parabola if . Consider that the standard equation of a parabola with vertex at... Click here to read more

The condition for a line to be a tangent to the parabola is that and the tangent to the parabola... Click here to read more

The equations of tangent and normal to the parabola at the point are and respectively. Consider that the standard equation... Click here to read more

Example: Find the equation of the tangent to the parabola parallel to the line . Solution: The given equation of a... Click here to read more

An ellipse is the set of all points, the sum of whose distances from two fixed points is a given... Click here to read more

To find the equation of an ellipse, let be any point of the ellipse and be the corresponding point on... Click here to read more

Example: Find the equation of the ellipse having center at origin, focus at and one vertex at the point . Since... Click here to read more

There are two types of ellipses: one ellipse has the X-axis as the major axis and the other has the Y-axis... Click here to read more

The length of the latus rectum of the ellipse is . The chord through the focus and perpendicular to the... Click here to read more

In geometry, a straight line is said to be the tangent to an ellipse it this straight line meets the... Click here to read more

The line intersects with the ellipse at two points maximum and the condition for such intersection is that . Consider... Click here to read more

The line does not intersect the ellipse, so the condition for no intersection is that . Consider that the... Click here to read more