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The Cartesian coordinates of P are those three numbers, in the chosen order. The reverse construction determines the point P given its three coordinates. Alternatively, each coordinate of a point P can be taken as the distance from P to the plane defined by the other two axes, with the sign determined by the orientation of the corresponding axis.
Carlyle circle of the quadratic equation x 2 − sx + p = 0. Given the quadratic equation x 2 − sx + p = 0. the circle in the coordinate plane having the line segment joining the points A(0, 1) and B(s, p) as a diameter is called the Carlyle circle of the quadratic equation. [1] [2] [3]
The sum of the squared lengths of any two chords intersecting at right angles at a given point is the same as that of any other two perpendicular chords intersecting at the same point and is given by 8r 2 − 4p 2, where r is the circle radius, and p is the distance from the centre point to the point of intersection. [14]
Let the x axis be the real axis and the axis be the imaginary axis. The position of the body can then be given as z {\displaystyle z} , a complex "vector": z = x + i y = R ( cos [ θ ( t ) ] + i sin [ θ ( t ) ] ) = R e i θ ( t ) , {\displaystyle z=x+iy=R\left(\cos[\theta (t)]+i\sin[\theta (t)]\right)=Re^{i\theta (t)}\,,} where i is ...
Thus a circle in the Euclidean plane was defined as the locus of a point that is at a given distance of a fixed point, the center of the circle. In modern mathematics, similar concepts are more frequently reformulated by describing shapes as sets; for instance, one says that the circle is the set of points that are at a given distance from the ...
A point is chosen as the pole and a ray from this point is taken as the polar axis. For a given angle θ, there is a single line through the pole whose angle with the polar axis is θ (measured counterclockwise from the axis to the line). Then there is a unique point on this line whose signed distance from the origin is r for given number r.
Conversely, it is easily shown that if a, b, c and d are constants and a, b, and c are not all zero, then the graph of the equation + + + =, is a plane having the vector = (,,) as a normal. [ citation needed ] This familiar equation for a plane is called the general form of the equation of the plane.
If P, T 1, T 2 lie on a common tangent, then P is the midpoint of ¯. In Euclidean geometry, the radical axis of two non-concentric circles is the set of points whose power with respect to the circles are equal. For this reason the radical axis is also called the power line or power bisector of the two circles. In detail: