Foci and Directrices

Find the focus and directrix of the parabola

Find the foci and directrices of the curve

Find the foci and directrices of the curve

A second-order curve passes through the origin, its focus is at the point $(-1,1)$, and the corresponding directrix has the equation $x+y-2=0$. Find the curve.

Find the ellipse passing through the point $(4,2)$ and having foci at the points $(4,3)$ and $(0,-1)$.

Find the ellipse with foci at the points $(1,2)$, $(2,1)$, passing through the point $(5,5)$.

Find the equilateral hyperbola from its directrix $x+y-1=0$ and focus $(1,1)$.

Find the parabola from its points $(1,1)$, $(1,2)$ and its directrix $x+y-1=0$.

A second-order curve passes through the point $(0,-1)$, has center $(1,1)$, and has as directrix the line $x+2y+1=0$. Find the curve.

Find the second-order curve from its points $(0,1)$, $(1,0)$, $(0,2)$ and focus $(1,1)$.

Find the second-order curve from its directrix $x+y+1=0$, axis $y=x$, and points $(0,0)$ and $(1,0)$.

Find the second-order curve from its points $(4,5)$, $(-3,4)$, focus $(1,1)$, and axis of symmetry $x+y-2=0$.

Find the parabola from its vertex $(0,0)$ and focus $(1,1)$.

Find the hyperbola from its asymptotes $x+y-1=0$, $x-y+1=0$ and focus $(0,2)$.

Find the locus of the foci of parabolas passing through the point $(1,-1)$ and having as directrix the line $x+y+1=0$.

Find the locus of the vertices of parabolas passing through a given point and having a given directrix.

Find the locus of the centers of equilateral hyperbolas passing through the point $(0,0)$ and having as directrix the line $x+y+1=0$.

What is the locus of the foci of second-order curves inscribed in a given parallelogram?