Learn Analytic Geometry with Step-by-Step Examples and Answers

Let ABCD be a convex quadrilateral, with $A C \cap B D = {O}$ , and $O B > O D$ , $O C > O A$
Let E and F be the midpoints of (AC) and (BD), $E F \cap A B = {J}$ , $E F \cap C D = {K}$ , $C J \cap B K = {L}$ . If M is the midpoint of (KJ), show that O-M-L are collinear.

lilmoore11p8 2022-07-10

How can I generate equations for 3D objects beyond the basics like spheres, cubes and pyramids? For example, how about a diamond?

Janet Forbes 2022-07-10

I want to find the general equation of the two lines of intersection of a one sheet hyperboloid to its tangent plane for the function
$F (x, y, z) = x^{2} + y^{2} - z^{2} = 1$
at
$(x_{0}, y_{0}, z_{0})$
The equation of the tangent plane is
$x_{0} x + y_{0} y - z_{0} z = 1$

Kolten Conrad 2022-07-08

Calculate angle between two vectors, given their rotation w.r.t. a third vector.
I have three vectors, $\vec{a}, \vec{b},$ and $\vec{c}$ in n-dimensional space. I know the coordinates of all three vectors and their dot products. Both $\vec{a}$ and $\vec{b}$ are rotated away from $\vec{c}$ by an angle $α$ , in their own respective directions, obtaining ${\vec{a}}^{'}$ and ${\vec{b}}^{'}$ . What is the angle between ${\vec{a}}^{'}$ and ${\vec{b}}^{'}$ ?

Sonia Ayers 2022-07-08

Let us start with these two equations of two lines:
$x + y = 4$
$x - y = 2$
They intersect at $(x, y) = (3, 1)$
Let us now translate (move) both lines so that they intersect at (0, 0). We need to move both lines by -3 along x-axis and by -1 along y-axis. So the equations of the lines become.
$(x + 3) + (y + 1) = 4$
$(x + 3) - (y + 1) = 2$
This is equivalent to
$x + y = 0$
$x - y = 0$
Why do the RHS become 0 for both equations? This happens no matter which two intersecting lines we begin with. What is the geometrical interpretation of this?

Ellen Chang 2022-07-07

Find the coordinates of $\vec{v}$ if $\vec{v} = 2 \vec{a} - 3 \vec{b} + 4 \vec{c}$ and $\vec{a} (4; 1), \vec{b} (1; 2)$ and $\vec{c} (2; 7)$

therightwomanwf 2022-07-03

What is the easiest way to see that the path
$\underline{r} : R \to R^{3} : t \mapsto (\sin t, \cos t, \cos t)$
traces out an ellipse in the plane $y = z$ ?

nidantasnu 2022-07-02

Find the equation of angle bisector passing through quadrant containing (2,3).
$L_{1} \Rightarrow 2 x + y - 1 = 0$
$L_{2} \Rightarrow 2 x - y + 3 = 0$

mistergoneo7 2022-07-02

I need to learn how to find the properties (foci, vertex, etc) of a conic given its equation, I know how to do it when the coeficient of $x y = 0$ , but I have troubles in the other case. Can someone recommend me a book, website o document where I can learn how to find the properties of a conic section given its equation please?

Dayanara Terry 2022-07-02

Find equation of circle passing through intersection of circle
$S = x^{2} + y^{2} - 12 x - 4 y - 10 = 0$
and
$L : 3 x + y = 10$ and having radius equal to that of circle S.

letumsnemesislh 2022-07-02

The points (2, -1, -2), (1, 3, 12), and (4, 2, 3) lie on a unique plane. Where does this plane cross the z-axis?

Aganippe76 2022-07-01

Two parallel lines with varying slopes and constant distance ( say 4 units) between them.
Let D be a line with equation $y = a x + b$ , with a a varying number.
So D has a changing slope and is rotating about point $P = (0, b)$ .

Savanah Boone 2022-07-01

Given the cube ABCD. EFGH. The point M is on the edge AD such that $| A M | = 2 | M D |$ . Calculate the tangent of the angle between the planes BCF and BGM.
A) $3 \sqrt{2}$
B) $2 \sqrt{2}$
C) $\frac{3}{2} \sqrt{2}$
D) $\frac{1}{2} \sqrt{2}$
E) $\sqrt{2}$

Abram Boyd 2022-06-27

Given a circle M of known center and radius and a right-angled triangle JKL with one vertex (J) at the circle's center and another vertex (K) at the circle's boundary, find the cartesian coordinates of the third vertex (L) (with respect to the center of the circle). The three sides of the triangle JKL are given, where one of the sides (A) is a radius of the circle. The coordinates for the vertex K are not known.

Karina Trujillo 2022-06-27

Let $△ A B C$ be a scalene triangle, I its incenter and G its centroid. Prove that the line GI intersects both (AB) and (AC) on the line segments (id est between the two points of extremities), if and only if there exists a $t \in [0; 1)$ for which we may write $a = t b + (1 - t) c$ , where a, b, c are the sides of the triangle in trigonometric notation (id est $a = B C$ , $b = A C$ , $c = A B$ ).

Micaela Simon 2022-06-27

Find the equation of an ellipse if its center is S(2, 1) and the edges of a triangle PQR are tangent lines to this ellipse. P(0, 0), Q(5, 0), R(0, 4).
My attempt: Let take a point on the line PQ. For example (m,0). Then we have an equation of a tangent line for this point: $(a_{11} m + a_{1}) x + (a_{12} m + a_{2}) y + (a_{1} m + a) = 0$ , where $a_{11}$ etc are coefficients of our ellipse: $a_{11} x^{2} + 2 a_{12} x y + a_{22} y^{2} + 2 a_{1} x + 2 a_{2} y + a = 0$ . Now if PQ: y = 0, then $(a_{11} m + a_{1}) = 0$ , $a_{12} m + a_{2} = 1$ , $a_{1} m + a = 0$ .I've tried this method for other 2 lines PR and RQ and I got 11 equations (including equations of a center)! Is there a better solution to this problem?

Bailee Short 2022-06-25

Rotate vector by a random little amount
Suppose there is a vector $v = (v_{x}, v_{y}, v_{z})$
I want to rotate this vector by a random little amount (let's say at most $10 º$ ). How can I do that?

Fletcher Hays 2022-06-24

I am making a simple 3D printed mechanism that consists of a tab and housing. Both the tab and housing have nubs that interface with each other, preventing the tab from exiting the housing. The nubs are circular arcs with height 0.5mm and length 3.00mm. Fusion 360 calculated this for me to 4 decimal places, but I want to know the formula I can use, given the height and nub length, in order to find the exact value for the distance. This is necessary because I need the formula in order to use the output value for other aspects of the design.

gnatopoditw 2022-06-23

Equation of the ellipse between 2 points (its vertices)
Start point and end point:
$A (x, y) B (x, y)$
Ellipse width:
$w = 2.5$
Midpoint or center of the ellipse:
$M = (\frac{A_{x} + B_{x}}{2}, \frac{A_{y} + B_{y}}{2})$
The angle (that would be the angle between points A and B):
$θ = ?$
Parametric equation of the ellipse with rotation:
$x = a * c o s (t) * c o s (θ) + b * s i n (t) * s i n (θ)$
$y = b * s i n (t) * c o s (θ) - a * c o s (t) * s i n (θ)$

excluderho 2022-06-22

Suppose, $2 x \cos α - 3 y \sin α = 6$ is equation of a variable straight line. From two point $A (\sqrt{5}, 0$ and $B (- \sqrt{5}, 0)$ foot of the altitude on that straight line is P and Q respectively. Show that the product of the length of two line segment AP and BQ is free of $α$ .

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Analytic Geometry Problems and Examples