Analytic geometry and conic sections

Given $A(-2,1)$ and $B(4,9)$, compute the distance $\overline{AB}$ and the midpoint of the segment $AB$.

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Write the equation of the line passing through $A(0,-1)$ and $B(2,3)$.

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Write the equation of the line passing through $P(3,-1)$ and perpendicular to the line of equation $x-2y+4=0$.

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Compute the distance from the point $P(1,-2)$ to the line of equation $4x+3y-10=0$.

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Find the vertex, the axis of symmetry and the intersections with the axes of the parabola

$$y=x^2-6x+5.$$

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Find the vertex, the concavity and the intersections with the $x$-axis of the parabola

$$y=-x^2+4x.$$

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Write, in standard form and in general form, the equation of the circle with center $C(-1,3)$ and radius $r=2$.

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Find the center and radius of the circle of equation

$$x^2+y^2-8x+6y=0.$$

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Find the semi-axes, vertices and foci of the ellipse of equation

$$\frac{x^2}{16}+\frac{y^2}{7}=1.$$

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Find the vertices, foci and asymptotes of the hyperbola of equation

$$\frac{x^2}{16}-\frac{y^2}{9}=1.$$

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Recognize the conic of equation

$$x^2+y^2+4x-2y+1=0$$

and find its elements.

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Recognize the conic of equation

$$4x^2-y^2-8x-4y-4=0,$$

reduce it to standard form and find its center, vertices and asymptotes.

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