Analytic Geometry Problems and Examples

When is a rectangle a regular polygon?

Find the distance d between the points ${\displaystyle P_1}$ and ${\displaystyle P_2}$
${\displaystyle P_1=(1.2,2.3)}$
${\displaystyle P_2=(-0.3,1.1)}$

A circle centered at (0, 0) includes the point A(0, 8). Point B lies on the circle such that the center angle with endpoints A and B measures 1 radian. What is the length of AB?

Find the area of the surface generaled by revolving the curve ${\displaystyle x=\frac{e^y+e^{-y}}{2}}$ in the interval ${\displaystyle 0\le y\le \ln 3}$ about the y-axis

Find the intersection of the line ${\displaystyle x=\frac{y-1}{2}=\frac{z-2}{3}}$ and the shere ${\displaystyle x^2+{(y-3)}^2+{(z-2)}^2=4}$
a) (1, 1, 0)
b) (1, 2, 3)
c) (0, 2, 3)
d) ${\displaystyle (\frac{1}{6},\frac{23}{6},\frac{17}{6})}$
e) ${\displaystyle (\frac{4}{7},\frac{15}{7},\frac{26}{7})}$

What is the volume of this triangular pyramid?

What is the volume of this cylinder?
Use ${\displaystyle \pi \approx 3.14}$ and round your answer to the nearest hundredth.

What is a hyperbola? What are the Cartesian equations for hyperbolas centered at the origin with foci on one of the coordinate axes? How can you find the foci, vertices, and directrices of such an ellipse from its equation?

Find ${\displaystyle m\mathrm{\angle }F}$

What is the surface area of this cone?
Use ${\displaystyle \pi \approx 3.14}$ and round your answer to the nearest hundredth.

What is the surface area of this cylinder?
Use ${\displaystyle \pi \approx 3.14}$ and round your answer to the nearest hundred/

Let E2 be the new system ${\displaystyle \{O,x,y\}}$ obtained by rotating the ${\displaystyle (O,x,y\}}$ perpendicular coordinate system arounf the point O in the Euclidean plane. Accordingly, the point ${\displaystyle \{O,\text{Whic of the following is the coordinate according to the},x,y\}}$ system?
Analytic geometry.

The lesson is analytic geometry standartize the conic
${\displaystyle (7x^2+4xy+4y^2-2x-6y+1=0}$

Write the equation of the circle with Center (-1, -8) and Radius: ${\displaystyle \sqrt{7}}$
a) ${\displaystyle x^2+2x+y^2+16y+58=0}$
b) ${\displaystyle x^2+2x+1+y^2+16y+64=7}$
c) ${\displaystyle x^2+2x+y^2+16y+72=0}$
d) ${\displaystyle x^2-2x+y^2-16y+58=0}$

a) Use Coordinate Geometry Computations to compute the intersection point of the lines AB and CD. The X and Y coordinates of stations A, B,C and D in feet are given as follows:
${\displaystyle X_A=8377.66Y_A=4340.92}$
${\displaystyle X_B=8719.43Y_B=4642.65}$
${\displaystyle X_C=84.36.76Y_C=4712.64}$
${\displaystyle X-\left\{D\right\}=8810.45Y_D=4325.72}$
b. Use an alternative method (e.g. analytical geometry) to compute the intersection point.

Let ${\displaystyle Q^2}$ be the rational plane of all ordered pairs (x,y) of rational numbers with the usual interpretations of the undefined geometric terms used in analytical geometry. Show that axiom C-1 and the elementary continuity principle fail in ${\displaystyle Q^2}$. (Hint: the segment from (0,0) to (1,1) can not be laid off on the x axis from the origin.
Axiom C-1: If A, B are two points on a line a, and if A' is a point upon the same or another line a′ , then, upon a given side of A′ on the straight line a′ , we can always find a point B′ so that the segment AB is congruent to the segment A′B′ . We indicate this relation by writing ${\displaystyle AB\stackrel{\sim }{=}A\prime B\prime }$. Every segment is congruent to itself; that is, we always have ${\displaystyle AB\stackrel{\sim }{=}AB}$.

A point moves so that the angle from the line joining it and the origin to the line (3, - 2) and (5, 7) is ${\displaystyle {45}^{\circ }}$ Find the equation of the locus.

Let ${\displaystyle v=zk}$ be the velocity field (in meters per second) of a fluid in R3. Calculate the flow rate (in cubic meters per second) through the upper hemispere ${\displaystyle \left(z>0\right)}$ of the sphere ${\displaystyle x^2+y^2+z^2=1}$

In the diagram below, a circle with centre O is given with diameter AB. The coordinates of A and B are (4 ; 4) and (2 ; -6) respectively. ya A(4; 4) B(2; -6)
a) Determine the coordinates of O, the centre of the circle.
b) Calculate the length of the radius of the circle.
c) Write down the equation of the circle in form:
${\displaystyle {(x-a)}^2+{(y-b)}^2=r^2}$
d) Write down the equation of the circle in form:
${\displaystyle a{x}^2+bx+c{y}^2+dy+e=0}$

Consider the sequence ${\displaystyle \left\{A_n\right\}}$, whose nth term is given by ${\displaystyle A_n={(1+\frac{2}{n})}^{3n}}$
Use limits to analyticlly show how we can determine whether the sequence is convergent or divergent. If it is convergent, find the limit of the sequence. Write a sentence to summariae your findings.

Find the area of the surface generaled by revolving the curve ${\displaystyle x=\frac{y^3}{5},0\le y\le 3}$ about the y-axis.

Find the Amplitude, Midline, Midline, period and Asymptotes for
${\displaystyle f\left(x\right)=-\cot \frac{1}{3}\left(x\right)+2}$