Identifying a Parabola Using Rotation of Axes
(a) Use rotation of axes to show that the following equation represents a parabola.
(b) Find the and coordinates of the vertex, and focus.
(c) Find the equation of the directrix in and coordinates.
The given equation represents a parabola using rotation of axes.
The given equation is,
The standard equation of a conic is,
Suppose the and axes in a coordinate plane are rotated through the acute angle to produce the and axes. Then the coordinates and of a point in the and plane are related as,
Compare the given equation with the standard equation to get,
On rotation of axes, the acute angle satisfies,
Substitute for and for in equations and
The and coordinates of the vertex, and focus.
The equation of the directrix in and coordinates
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