###### Answer: c. (0 -3)

Thu Jan 03 2008 13:30:00 GMT-0500 (Eastern Standard Time) · Parabolas with center of inversion at the vertex . The equation of a parabola is up to … x = y 2. In polar coordinates this becomes = . The inverse curve then has equation = = which is the cissoid of Diocles. Conic sections with center of inversion at a focus . The polar equation of a conic section with one focus at the origin is up to similarity = + where e is …

If a parabola is positioned in Cartesian coordinates with its vertex at the origin and its axis of symmetry along the y -axis so the parabola opens upward its equation is = where is its focal length. (See ” Parabola #In a cartesian coordinate system”.)

At the vertex () = [] the radius of curvature equals R(0) = 0.5 (see figure). The parabola has fourth order contact with its osculating circle there. For large t the radius of curvature increases ~ t 3 that is the curve straightens more and more. Lissajous curve

Orthoptic of a parabola . Any parabola can be transformed by a rigid motion (angles are not changed) into a parabola with equation =.The slope at a point of the parabola is =.Replacing gives the parametric representation of the parabola with the tangent slope as parameter: ( ). The tangent has the equation = + with the still unknown which can be determined by inserting the coordinates of the …

After introducing Cartesian coordinates the focus – directrix property can be used to produce the equations satisfied by the points of the conic section. By means of a change of coordinates (rotation and translation of axes) t…