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Show that the cone whose vertex is at the origin and which passes through the curve vif intersection of the sphere x^2+y^2+z^2=3p^2 and any plane which is at the distance‘p' from the origin has mutually perpendicular generator s.
Let A be the area of a triangle with semi- perimeter p. Show that A cannot be greater than p^3/3√3.
Denote by a, b and c the column vectors a = (1 2 3), b = (-2 1 -3), c = (-2 -1 1) Calculate 2a - 5b, 2a- 5b +c, a'.b,
The plane making intercept 1 at the z-axis and parallel to the xy- plane intersects the cone x^2+y^2=z^2tan^2 ϑ in a circle.
True or false with full explanation
Show that the point (0,3+√5) lies on the ellipse with foci (2,3) and (-2,3) and semi major axis 3.
Show that the conicoid x^2+2y^2+2yz+2x+4y+8z+1=0 is central. Find the new equation of the conicoid if the origin is shifted to its centre.
To find a missing side length set up and solve a ____________________. Put the measurements of the smaller figure on top and the bigger figure on the bottom.

An archeologist found the remains of an ancient wheel, which she then placed on a grid. If an arc of the wheel passes through A(-7, 0), B(-3, 4) and C(7, 0) locate the center of the wheel, and the standard equation of the circle defining its boundary.

An engineer is working on making a diagram for the cooling tower he was going to construct in the future. Cooling towers can be seen as towers with inward curving sides, or simply, hyperbola looking sides. His daughter asks him how high the center of the hyperbolic sides of tower was y=±(24/7)x+15 and the answers that the asymptotes of the hyperbola are the following, and the area of itsa auxilliary rectangle is 168 squats units. The daughter then find the foci of the hyperbola. At what points on the cartesian plane are these foci found
A big room is constructed so that the ceiling is a dome that is semielliptical in shape. If a person stands at one focus and speaks, the sound that is made bounces off the ceiling and gets reflected to the other focus. Thus, if two people stand at the foci (ignoring their heights), they will be able to hear each other. If the room is 34 m long and 8 m high, how far from the center should each of two people stand if they would like to whisper back and forth and hear each other?
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