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Questions tagged [analytic-geometry]

Questions on the use of algebraic techniques for proving geometric facts. Analytic Geometry is a branch of algebra that is used to model geometric objects - points, (straight) lines, and circles being the most basic of these. It is concerned with defining and representing geometrical shapes in a numerical way.

Analytic geometry, also called coordinate geometry, mathematical subject in which algebraic symbolism and methods are used to represent and solve problems in geometry.

The importance of analytic geometry is that it establishes a correspondence between geometric curves and algebraic equations. This correspondence makes it possible to reformulate problems in geometry as equivalent problems in algebra, and vice versa; the methods of either subject can then be used to solve problems in the other.

Analytic geometry was introduced by René Descartes in $1637$ and was of fundamental importance in the development of the calculus by Sir Isaac Newton and G. W. Leibniz in the late $17^{th}$ cent. More recently it has served as the basis for the modern development and exploitation of algebraic geometry.

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Coordinate geometry triangle problem

Find the equations which all points inside the triangle formed by the points $(1,3)$, $(5,0)$ and $(-1,2)$ must satisfy Now the answer mentioned is $$3x+2y \ge 0$$$$2x+y-13 \ge0$$$$2x-3y-12 \le0$$ Now I found the equation of the triangular sides and…
Harsh Sharma
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A question on analytic geometry

A point $Q(x_2,y_2)$ is on the line segment passing through $R(-2,5)$ and $S(4,1)$. Find the coordinates of $Q$ if it is twice as far from $R$ as from $S$.
jjjj
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Finding straight lines

I am new to analytical geometry and excuse me for my notations. We have four lines: $l_1: u_1x + v_1y + r_1 = 0$ $l_2: u_2x + v_2y + r_2 = 0$ $m_1: u_3x + v_3y + r_3 = 0$ $m_2: u_4x + v_4y + r_4 = 0$ We know that $l_1$ and $l_2$ cross int $(l_1,…
murloc
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Generating an Equation for Quadratic Bezier Curve

So I know that a standard equation for a quadratic Bezier Curve is: $C(t)=(1-t)^2P_0+2t(1-t)P_1+t^2P_2$ I have been asked to generate an equation to model my quadratic bezier curve. I known that my $t=0.5$ and my control points are:…
Emma
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Find the coordinates of P and Q.

A line is drawn through the point $A(1,2)$ to cut the line $2y=3x-5$ in $P$ and the line $x+y=12$ in $Q$. If $AQ=2AP$, find the coordinates of $P$ and $Q$. From: Mathematics, The Core Course for A-level, Bostock and Chandler. Chapter 4 Q15. The…
Kantura
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Using cross product prove that if $\vec{u} \times \vec{v} = \vec{0}$ and $\vec{u} \cdot \vec{v} = 0$ then $\vec{v} = \vec{0}$

I am asked to elaborate on the following proof: Let $\vec{u} \neq \vec{0}$. Prove that if $\vec{u} \times \vec{v} = \vec{0}$ and $\vec{u} \cdot \vec{v} = 0$ then $\vec{v} = \vec{0}$. My attempt was to say that $$ u . v = 0 \Rightarrow (u_1, u_2,…
bru1987
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Prove that locus of orthocentre is $x+y=0$

The base of a triangle is axis of $x$ and its other $2$ sides are given by the equations :$y = (1+α)\frac{x}{α} + (1+α)$ and $y = (1+β) \frac{x}{β} + (1+β)$.Prove that the locus of its orthocentre is the line $x+y = 0 $ Finding orthocentre by…
MathGeek
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Two points are given as A(2,0) and B(8,0). What's the value of y (y>0), so that C(0,y) is such that angle ACB has maximum value?

My first guess is that it could be found as first derivative of some function, but I don't have idea what that function could be.
Nafel Hekun
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What are the coordinates of the intersection points of two circles?

You have 2 circles that intersect in 2 points. You know the coordinates of their centers and you also know their radius. My question is: What are the coordinates of these 2 intersection points?
Theo Cristea
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Lines tangent to two circles

I'm trying to find the lines tangent to two circles. I've seen several examples but with poorlyy explained methods. Given the circle $(x-x_{0})^2+(y-y_{0})^2=r_{1}^2$ and the the line equation $y=ax+b$ one of the method is based on the…
Gabu
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Show that pair of straight lines $ax^{2}+2hxy+ay^{2}+2gx+2fy+c=0$ meet coordinate axes in concyclic points. Also find equation of the circle

Show that pair of straight lines $ax^{2}+2hxy+ay^{2}+2gx+2fy+c=0$ meet coordinate axes in concyclic points. Also find equation of the circle through those cyclic points My Attempt: Given equation to the pair of straight lines…
user335710
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Condition for three points to lie on a Sphere?

If A and B are the points (3,4,5) and (-1,3,-7) respectively then the set of the points P such that $PA^2+PB^2 = K^2$ where K is a constant lie on a proper sphere if K = 1 or K^2 <>= 161/2? The correct answer is K^2 > 161/2. How? I know that three…
Matt
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Suppose you graphed every single point of the form (2t + 3, 3-3t).

Suppose you graphed every single point of the form $(2t + 3, 3-3t)$. For example, when $t=2$, we have $2t + 3 = 7$ and $3-3t = -3$, so $(7,-3)$ is on the graph. Explain why the graph is a line, and find an equation whose graph is this line.
Marcie
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Show that the three distinct points $(p,p^2)$, $(q,q^2)$ and $(r,r^2)$ can never be collinear.

Show that the three distinct points $(p,p^2)$, $(q,q^2)$ and $(r,r^2)$ can never be collinear. I can think of the graph $y=x^2$ to solve the above problem graphically. However, I wanted to solve it mathematically, so I tried to find the area of the…
Swadhin
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When does $ax+by+c=0$ represents a family of straight lines passing through a fixed point?

a first degree linear equation $ax+by+c=0$ represents a family of straight lines passing through a fixed point if and only if there is linear relationship between a,b and c? How can we prove this? Can the relation between a,b,c be a quadratic one…
Matt
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