Are the following lines k and l conjugate imaginary lines?
line k has line coordinates (1-i,i,4) line l has line coordinates (2,-1-i,4-4i) |
Calculate three imaginary points on the line x - 2 y + z = 0 |
Calculate the real point on the line k(2 + i, 1, -i) |
Level 2 problems
Calculate the real values m and n such that the point
(2 - i, 3 - n i, m + i)is a real point. |
Calculate the components of the curve
x^{2} + 4 x - 6 = 0 |
Calculate the components of the curve
x^{2} + 4 x y + 3 y^{2} - 2 x z - 4 y z + z^{2}= 0 |
Calculate the real values of m such that the following conic section
is degenerated.
x^{2} - 4 x y + 2 y^{2} - 5 y - m x + 2 = 0 |
Calculate the double points of the following conic section
x^{2} + 4 x y + 3 y^{2} - 2 x z - 4 y z + z^{2}= 0 |
Calculate the double points of the following conic section
x^{2} + 2 x y + y^{2} - 8 x z - 8 y z + 16 z^{2}= 0 |
Calculate the tangent line in point P(2,0) of the conic section x^{2} - 4 x y - y^{2} + 2 x - 4 y - 8 = 0 |
Calculate the tangent line in point P(2,0) of the conic section x^{2} + x y - 6 y^{2} - 4 x z - 2 y z + 4 z^{2}= 0 |
The tangent line in point P(?,?) of the conic section x^{2} - 4 x y - y^{2} + 2 x - 4 y - 8 = 0 is x - 2 y - 2 = 0 Calculate the coordinates of the tangent point. |
Calculate the asymptotes of the conic section x^{2} - 4 x y - y^{2} + 2 x - 4 y - 8 = 0 |
Calculate the asymptotes of the conic section x^{2} + 2 x y + y^{2} - 4 x - 5 y + 7 = 0 |
Calculate the asymptotes of the conic section x^{2} + 2 x y - 4 x = 0 |
Calculate the asymptotes of the conic section x^{2}- 2 x = 0 |
Search the equation of the conic section with asymptotes x = 0 and x + 2 y - 41 = 0 and through the point P(2,1). |
Calculate the equation of the system of conic sections with basic points A(1,2); B(2,0); C(-1,1); D(0,3). |
Calculate the equation of the system of conic sections with basic points A(1,2); B(2,0); C(-1,1); C(-1,1) and such that the line c with equation x + y = 0 is the tangent line in point C. |
Calculate the equation of the system of conic sections with basic points A(1,2); A(1,2); B(2,0); B(2,0) and such that the line a with equation x + y - 3 z = 0 is the tangent line in point A and b with equation x + 2 y - 2 z = 0 is the tangent line in point B. |
Calculate the basic points of the system with basic conic sections x^{2} + 2 x y + 7 y^{2} - 5 x z - 17 y z + 6 z^{2}= 0 and -3 x^{2} - 4 x y + 5 y^{2} + 3 x z - 9 y z + 6 z^{2}= 0 |
Calculate the polar line of P(1,1,1) relative to the conic section -3 x^{2} - 4 x y + 5 y^{2} + 3 x z - 9 y z + 6 z^{2}= 0 |
Calculate the polar line of P(1,1,1) relative to the conic section x^{2} - y^{2} - 2 x z + 2 y z = 0 |
Calculate the polar line p of P(1,0,1) relative to the conic section (x - y) (x + y - 2 z) = 0This conic section has double point S(1,1,1). Show that the components of the conic section, the line SP and the line p form a harmonic quartet of lines. |
Calculate the point C of a polar triangle ABC of the conic section x^{2} + 2 x y + 7 y^{2} - 5 x z - 17 y z + 6 z^{2}= 0 if you know that A(2,1,1) and B(0,15,1). |
The lines a,b and c are the polar lines of the vertices of a triangle ABC,
relative to a not degenerated conic section K.
P is the intersection point of a and BC. Prove that P,Q and R are collinear. |
Calculate the center-point of the conic section
x^{2} + 2 x y + 7 y^{2} - 5 x z - 17 y z + 6 z^{2}= 0 |
Calculate the center-point of the conic section
x^{2} + 4 x y + 4 y^{2} + 2 x z + 4 y z - 8 z^{2} = 0 |
What is the general equation of a conic section with the point (0,0,1) as a center-point. |
Calculate the center-line of the conic section x^{2} + 2 x y + 7 y^{2} - 5 x z - 17 y z + 6 z^{2}= 0 conjugated to the direction with slope -1. |
Calculate the value of r such that the line x + 2 y + z = 0 is a center-line of the conic section x^{2} + 2 x y - y^{2} - r x z + r y z = 0 |
Calculate the value of r and s such that the line x + 2 y + z = 0 is a center-line conjugated to the direction with slope -2 relative to the conic section x^{2} + 2 x y - y^{2} - r x z + s y z -2 z^{2} = 0 |
Calculate the direction conjugated to (1,-2,0) relative to the conic section x^{2} + 2 x y - y^{2} - 4 x z + 2 y z -2 z^{2} = 0 |
Calculate the center-line conjugated to the center-line 4 x - 2 y + z = 0 relative to the conic section x^{2} + 2 x y - y^{2} - 4 x z + 2 y z -2 z^{2} = 0 |
Calculate the axes and the vertices of the conic section x^{2} + 2 x y - (1/2) y^{2} - 4 x z + 2 y z -6 z^{2} = 0 |
Calculate the axis of the parabola x^{2} + 2 x y + y^{2} - 4 x + 2 y - 6 = 0 |
Calculate the focus and the directrix of the parabola x^{2} + 2 x y + y^{2} - 4 x + 2 y - 6 = 0 |
The origin point (0,0,1) is the focus of a not degenerated parabola. The distance from the origin to the vertex of the parabola is r. Show that the distance from the origin to the directrix is 2r. |
Given is a quadratic equation in z with parameters x and y.
z^{2} - x z + (x - y)^{2} = 0The parameters x and y are the coordinates of a point P relative to an orthonormal coordinate system in a plane. Calculate the locus of point P such that the quadratic equation has equal roots. |
A variable circle c has equation
x^{2} + y^{2} - 2 (t^{2} - 3 t + 1) x - 2 (t^{2} + 2 t) y + t = 0The number t is a parameter. Calculate the locus of the center of the circle. |
We have, in an orthonormal coordinate system,
a circle c through the origin O and with a fixed radius R.
Now, the circle c starts to rotate about the origin O.
From a fixed point at infinity, we draw tangent lines at the rotating circle.
Calculate the locus of all points of tangency. |
Calculate the locus of a point P such that the tangent lines from P to the ellipse b^{2}x^{2} + a^{2}y^{2} - a^{2}b^{2} = 0 are orthogonal lines. |
The conic section with equation m x^{2} + 2 xy + m y^{2} + 2m x - 2m y = 0 is variable because of the parameter m. Find the locus of the variable center-point. |