## Parametric Equations of a Circle

Draw a circle with centre at $$O\left( {0,0} \right)$$ and with a radius equal to $$r$$ which is the fixed… Click here to read more

From basic to higher mathematics

Draw a circle with centre at $$O\left( {0,0} \right)$$ and with a radius equal to $$r$$ which is the fixed… Click here to read more

Consider the equation of a circle in general form is \[\boxed{{x^2} + {y^2} + 2gx + 2fy + c =… Click here to read more

Let $$A\left( {{x_1},{y_1}} \right)$$ and $$B\left( {{x_2},{y_2}} \right)$$ be the end points of the diameter of the circle as shown… Click here to read more

Consider the general equation a circle is given by \[{x^2} + {y^2} + 2gx + 2fy + c = 0\]… Click here to read more

Consider the general equation a circle is given by \[{x^2} + {y^2} + 2gx + 2fy + c = 0\,\,\,{\text{… 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 $$\left( {0,0} \right)$$ and radius $$r$$. Then the equation of this… Click here to read more

Consider the equation of a circle with center at origin $$\left( {0,0} \right)$$ and radius $$r$$. Then the equation of… 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 $${x^2} + {y^2} + 2gx + 2fy + c = 0$$… Click here to read more

The point $$P\left( {{x_1},{y_1}} \right)$$ lies outside, on or inside the circle $${x^2} + {y^2} + 2gx + 2fy +… Click here to read more

Two tangents can be drawn to a circle $${x^2} + {y^2} = {r^2}$$ from any point $$P\left( {{x_1},{y_1}} \right)$$. The… Click here to read more

We will find the equation of tangent lines drawn from the point $$\left( {1,5} \right)$$ to the circle given by \[{x^2}… Click here to read more

Let the tangent drawn from the point $$P\left( {{x_1},{y_1}} \right)$$ meet the circle at the point $$T$$ as shown in… Click here to read more

The length of the diameter of the circle $${x^2} + {y^2} = {r^2}$$ is equal to $$2r$$. Consider the equation… 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 \[{x^2}… 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 $$O\left( {0,0} \right)$$ is given by \[{x^2} +… 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 $${y^2} = 4ax$$ the parabola opens to the right if $$a > 0$$ and… Click here to read more

The point of parabola closed to focus is the vertex. Let the given parabola be \[{y^2} = 4ax\,\,\,\,{\text{ – –… Click here to read more

Example 1: Find an equation of the parabola having its focus at $$\left( {0, – 3} \right)$$ and as its… 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 $$y = mx + c$$ intersects the parabola $${y^2} = 4ax$$ at two points maximum and the condition… Click here to read more

The line $$y = mx + c$$ does not intersect the parabola $${y^2} = 4ax$$ if $$a < mc$$. Consider… Click here to read more

The condition for a line $$y = mx + c$$ to be a tangent to the parabola $${y^2} = 4ax$$… Click here to read more

The equations of tangent and normal to the parabola $${y^2} = 4ax$$ at the point $$\left( {{x_1},{y_1}} \right)$$ are $${y_1}y… Click here to read more

Example: Find the equation of the tangent to the parabola $${y^2} = 13x$$ parallel to the line $$7x – 9y +… 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 $$P\left( {x,y} \right)$$ be any point of the ellipse and $$M\left( {\frac{a}{e},y}… Click here to read more

Example: Find the equation of the ellipse having center at origin, focus at $$\left( {3,0} \right)$$ and one vertex at the… 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 $$\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1,\,\,a > b$$ is $$\frac{{2{b^2}}}{a}$$. The chord… 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 $$y = mx + c$$ intersects with the ellipse $$\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$$ at two points maximum… Click here to read more

The line $$y = mx + c$$ does not intersect the ellipse $$\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$$, so the condition… Click here to read more

The condition for a line $$y = mx + c$$ to be the tangent to the ellipse $$\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}}… Click here to read more

The equations of tangent and normal to the ellipse $$\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$$ at the point $$\left( {{x_1},{y_1}} \right)$$… Click here to read more

Find the equation of the tangent and normal to the ellipse $$\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$$ at the point $$\left(… Click here to read more

Example 1: An arch is in the form of a semi-ellipse, and it is $$48$$ feet wide at the base… Click here to read more

A hyperbola is the set of all points in the plane, the difference of whose distance from two fixed distinct… Click here to read more