Shifting of Origin in Hyperbola

A circle is built around a point called the centre. It is impossible to draw a circle without a centre. If a particular circular figure does not have a defined centre, we cannot consider it a circle.

The same goes for a hyperbola. The centre of a circle is the middle point of a straight line joining two opposite points on its circumference. The hyperbola’s centre is the middle point of the line joining the two foci of the hyperbola.

Definitions

Hyperbola

Hyperbola is a particular shape that is part of a group of shapes called conics. Conics are shapes that are a part of the geometric figure- cone. When we cut a cone at different angles, we get curved shapes. These shapes are called conics. There are four conics — circle, ellipse, parabola, and hyperbola.

The imaginary line which divides the hyperbola (any conic section) into symmetrical figures is called the axis.

Foci are two assumed points around which the hyperbola curve is constructed.

The axis which passes through the centre and the two foci is called the transverse axis, and the line which passes through the centre and is perpendicular to the transverse axis is called the Conjugate axis.

Vertices are the points where the curves intersect the transverse axis.

Asymptotes are two straight lines that intersect at the centre of the hyperbola and run along the branches of the curve to intersect the curve at infinity.

Equations

In mathematics, these shapes are represented using algebraic equations. The hyperbola formula/equation for hyperbola isx^2/a² -y^2/b² = 1; for when the x-axis is the transverse axis (horizontal hyperbola).

And x^2/a² -y^2/b²= -1; for when the y-axis is the transverse axis (vertical hyperbola).

Here x = x coordinate of any assumed point P on the hyperbola and y = y coordinate of any assumed point P on the hyperbola.

Shifting the origin of a hyperbola

A standard hyperbola is drawn with its centre/origin coinciding with the centre of the coordinate system O(0,0).

Shifting of origin/centre means the hyperbola’s centre is moved to another point on the coordinate plane, i.e., the x-y plane.

The equation for a shifted hyperbola is,(x-h)^2/a² – (y-k)^2/2b² =1, where O’(h,k) is the new centre of the hyperbola.

h & k is subtracted from x & y so that the new X & Y values are different from the existing values. This change in x & y results causes a shift in the centre of the hyperbola without changing the size of the curve.

Hence, the centre shifts by h units to the right/left and shifts k units up/down. (depends on the sign of values h and k)

By shifting the centre, we change the location of the hyperbola. At the new site, every point on the hyperbola has a new set of coordinates compared to the previous location.