Circle
An Analysis of a Circle and it's relationship with tangents and straight lines
Contents
- Definition
- The General Equation Of A Circle
- The Equation Of A Circle Through Three Given Points
- The Equation To A Tangent Of A Circle At A Given Point
- The Length Of The Tangent From A Given External Point To A Circle
- Orthogonal Circles
- The Points Of Intersection Of A Straight Line And A Circle
- Page Comments
Definition
A circle is a simple shape of Euclidean geometry consisting of those points in a plane which are equidistant from a given point, called the centre.
A circle's diameter is the length of a line segment whose endpoints lie on the circle and which passes through the centre.
The radius is half the diameter of the circle.
A circle's diameter is the length of a line segment whose endpoints lie on the circle and which passes through the centre.
The radius is half the diameter of the circle.
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Therefore
If then the equation becomes that of a circle radius R centred on the origin. This is an equation in the second degree in which the coefficients of and are equal and the term is missing.</p>
The General Equation Of A Circle
is a mathematical constant whose value is the ratio of any Euclidean plane circle's circumference to its diameter.
The equation of a circle can be written as :
The equation of the circle is
This can be rearranged as:
From which it can be seen that the centre is and the radius is
How To Find The Equation Of The Circle Whose Diameter Is The Join Of The Points A And B
- Let the coordinates of A and B be and
From the diagram it can be seen that since is a right angle the angle is a right angle. The slopes of and are respectively given by:
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andSince AP and PB are perpendicular, the products of their slopes is :This relationship is satisfied by the coordinates of any point P on the Circle and is therefore the required equation to the circle whose diameter joins the points A and B
The Equation Of A Circle Through Three Given Points
Suppose that the three given points are Substituting these values for and in the General Equation of a Circle: Then and These three equations are sufficient to enable the constants and hence the equation of the Circle to be determined.To Find The Equation To The Tangent Of Gradient M To The Circle With Centre The Origin.
For a circle of radius r:
- the area
- the circumference
The equation of the circle is
Suppose the equation of the tangent is . The length of the perpendicular from the tangent to the centre of the circle must be 'a'
The tangent of gradient m if therefore given by:
The Equation To A Tangent Of A Circle At A Given Point
Suppose that we require the equation of the tangent at the point to the circle. Differentiating with respect to x Therefore the gradient at the point is given by: The equation of tangent through the point on the circle with slope equal to the gradient of the curve is: This can be written as: But since the point lies on the circle we can make the following substitution: byHence the required equation can be written as:
The Length Of The Tangent From A Given External Point To A Circle
For each tangent to a circle we can find a radius such as the tangent is perpendicular to it.
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Thus the square of the length of the tangent drawn to the circle from the point is obtained by writing a for and for in the left hand side of the equation of a Circle.
Orthogonal Circles
Two circles are said to be orthogonal when the tangents at their points of intersection are at right angles.
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The Points Of Intersection Of A Straight Line And A Circle
A tangent to a circle is a straight line that touches the circle at a single point
A secant is a straight line cutting the circle at two points.
A secant is a straight line cutting the circle at two points.
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The Square of the length of the chord