Questions tagged [tangent-line]

For questions on the tangent line, the unique straight line that is the best linear approximation to a function at a point.

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Competition problem based on triangles [closed]

Let $ABC$ be a triangle with $AC > AB$, and denote its circumcircle by $\Omega$ and incentre by $I$. Let its incircle meet sides $BC, CA, AB$ at $D, E, F$ respectively. Let $X$ and $Y$ be two ...
stephan's user avatar
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ODE has many tangent solutions

This is perhaps one of the most over-asked questions on this site: say here, or here. However, the answers are not satisfactory, especially of the second assertion. The first assertion is ...
Kadmos's user avatar
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Find the equation of tangent through the point (3,4) on the circle $x^2+y^2 = 9$

radius = 3 Let $y-y_1 = m(x-x_1)$ be the equation of tangent. Since, tangent passes through (3,4) $or,\text{ }y - 4 = m(x - 3)$ $or,\text{ }y - 4 = mx - 3m$ $or,\text{ }mx +y - 3m + 4 = 0$ Thus, ...
Abhishek Kharel's user avatar
2 votes
4 answers
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Is there a more geometrical definition of the tangent line of a curve? based on the intuitive idea that a tangent line only touches at one point

I asked this question to my calculus teacher and it was a frustrating experience, basically he would say, over and over again, that a tangent line at a point is a line that goes through that point and ...
zlaaemi's user avatar
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Finding the radius of the circle that is externally tangential to two given circles and the $x$-axis, without trigonometry [duplicate]

Given two circles: $\color{red}{\Gamma_1: x^2+y^2=1}$ $\color{blue}{\Gamma_2: x^2+(y+\frac{1}{2})^2=\frac{1}{4}}$ $\color{green}{\Gamma_3: \dots ?}$ where $\color{green}{\Gamma_3}$ touchs $\color{red}{...
Hussain-Alqatari's user avatar
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Torsion for a 3D curve can be positive or negative, while curvature is always taken to be positive. Why?

$$\frac{dB}{dS} = -\tau N = \tau(-N)$$ $$\frac{dT}{dS} = \kappa N$$ where $\tau$ is the torsion and $\kappa$ is the curvature. $\frac{dT}{dS}$, by convention, is defined to be in the direction of $N$, ...
Sasikuttan's user avatar
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Geometric question on finding the length of the tangent

Consider a circle $ C $ with radius $ r $ and a point $P $ outside the circle. Construct two tangents from $P$ to the circle, touching the circle at points $ A $ and $ B $. Let $ O $ be the center ...
StudyME's user avatar
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Finding perpendicular line at on a curve given only three points

I am working on my thesis and I am out of my depth with the mathematical formula for defining the slope of a perpendicular line for a curve through three points. I do not know anything besides 3 XY ...
OlliM's user avatar
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Extrema of derivate are where tangent crosses the curve.

In this article https://www.jstor.org/stable/2310782 i found this proposition: Let $f$ be a differentiable function defined on an open interval $(a, b)$ containing the point $x_0$. Let: (B) There ...
user791759's user avatar
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1 answer
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Find the normal vector or an ellipsoid given the scale vector?

Each point of a unit sphere is conveniently also its own normal. I am starting with a unit sphere and then multiplying each point by a scale vector to create an ellipsoid. I would like to know the ...
Lorry Laurence mcLarry's user avatar
1 vote
0 answers
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Does the Tangent theorem for circumferences hold in the sphere $S^2$? [closed]

I am wondering if a kind of theorem of tangents from a point to a circumference holds in the sphere. The situation that I have in mind is the following. Take two geodesics $\gamma_1$ and $\gamma_2$ ...
Cris's user avatar
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1 answer
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Find the equation of the tangent to a system using implicit function theorem

Here is my system of equations: $C: \begin{cases}x^2 + y^2 +z^2 = 14\\ x^3+y^3+z^3=36 \end{cases}$ Firstly I managed to show that for all $a \in C$, the implicit function theorem applies to express $...
Alex's user avatar
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Calculating angle for line tangent to circle through a point

I have a circle of fixed radius $r$. I have a target that is $x$ units laterally separated from the center of the circle, and $y$ units vertically. I need to calculate the angle $θ$ which is the ...
Phrogz's user avatar
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Given a circle and an external point, find the x intercept of the line tangent to the circle and goes through the point.

The equation of the circle is given by $x^2+(y-r)^2=r^2$ where $r$ is the radius. The point is located at the point $(d,h)$. Here is my approach to this: the general equation of all lines that passes ...
Raymond Li's user avatar
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1 answer
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tangential angle of bifoliate

Consider the curve given by polar equation$$r=f(\theta)={8\cos\theta\sin^2\theta\over3+\cos(4\theta)}$$for $\theta$ in $[0,\pi]$. By Mathworld's equation (9) the tangential angle is given by $$\phi(θ) ...
hbghlyj's user avatar
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3 votes
1 answer
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How to construct a parabola that tangent to the function $\max \left\{ {x,0} \right\}$ and lie above it?

What would be the correct way to construct a parabola that tangent to the function $\max \left\{ {x,0} \right\}$ and lies above it? One of the thing that is difficult is that the max function is ...
Tuong Nguyen Minh's user avatar
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Is the parabola the only differentiable convex even curve, such that all vertical lines, reflected on the tangent of the curve, converge into a point?

It is well known in optics that a parabola $y = x^2$ has this very important property for applications: $(P)$ The reflection of all vertical lines on the tangent of the parabola all converge into a ...
Basj's user avatar
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Why does it seem like that a tangent line of an odd-degree polynomial function crosses the curve at more than one point?

This question has bothered me for a long time. I know that a tangent line only crosses a curve at one specific point. However, consider this: Let $f(x)=x^3$ The derivative of this function is $f^\...
Napoleon Bonaparte's user avatar
5 votes
0 answers
152 views

A notion of "differentiation" based on secant rather than tangent

Given a differentiable real function $f$, the derivative $f'(x)$ is the slope of the tangent to the graph of $f$ at $(x,f(x))$. Suppose that, instead of the tangent, we look at the secant to the graph ...
Erel Segal-Halevi's user avatar
2 votes
0 answers
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Proving an identity of distances about tangent of a locus similar to conchoid

Let $l$ be a line and $A$ be a fixed point. Draw a line through $A$ meeting $l$ at $B$. Take the point $C$ on the half-line $BA$ such that $BC$ equals a given constant. Draw the locus of $C$ (called ...
hbghlyj's user avatar
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Circles formed by points of common tangency of two circles

I was playing around in Geogebra with circles and their common tangents. For any two random circles (yellow), I noticed that the four points of tangency of direct common tangents form a circle, and so ...
algorhythm's user avatar
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1 answer
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Circles1 with maximum and minimum values [closed]

I have a question regarding the maximum and minimum values of $y-3x+4$ for which point $(x,y)$ moves along circle $(x-2)^2+y^2=1$. I can calculate the question but I do not know where to locate the ...
Fawad's user avatar
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3 votes
1 answer
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Doubt regarding shortest distance between exponential and logarithmic curve?

Consider the two functions functions : $e^x$ and $\ln x$. I know that the shortest distance is along the common normal. But my teacher said that "both the curves are inverse of each other and ...
An_Elephant's user avatar
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How to find the equation of the line tangent to a curve in a given point [duplicate]

Let $f$ be: $$f(x) = 100 - x^2 $$ As far as I understand, the slope of the line tangent to such curve is given by its derivative: $$ f'(x) = -2x $$ I want the equation of the line tangent to $f$ at ...
Dan's user avatar
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Implicit Differentiation: Proving a point is perpendicular at a certain point on two interesecting tangents.

Given the following question: Consider the curves C1 and C2 defined as follows; C1 : xy = 4 , x > 0 C2 : y^2 - x^2 = 2 , x > 0 Let P(a,b) be a unique point where the curves C1 and C2 intersect. ...
Liam Gannaway's user avatar
3 votes
2 answers
60 views

How do I calculate the intersection points between inverse shared tangent lines, a circle, and a bounding box?

Context I am attempting to draw a polygon with 4 or 5 corners using a software library that draws shapes by taking in an array of corners in the form of $(X,Y)$. I am somewhat woefully under skilled ...
Suni's user avatar
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6 votes
4 answers
639 views

Typo in book or I am wrong?

Find the equation of the tangent to the parabola $y = x^2$, if the $x$-intercept of the tangent is $2$. Now $$y = mx + b$$ $$0 = m(2) + 2$$ $$m = -1$$ so $$m = \frac{y-0}{x-2}$$ $$m(x-2) = y$$ $$-x-y =...
MrJonesBones's user avatar
-1 votes
2 answers
297 views

how do i find the equations of L1 and L2 [closed]

L1 and L2 are perpendicular. the equation of the circle is given as $x^2+6x+y^2-2y=7$. line L1 cuts the circle at $P$. L2 cuts the circle at $Q$. I need to find the equations of the lines L1 and L2. I ...
jggf65's user avatar
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5 votes
1 answer
213 views

Constant area of triangle from tangent line and axes

Let $f:(0,+\infty)\to (0,+\infty)$ be a differentiable function such that $f'(x)\not=0$ with the following property: the area of the triangle formed by the tangent line of $C_f$ at a point $M(x_0,f(...
1123581321's user avatar
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0 votes
0 answers
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Calculating unit vectors B , T , N a (probably ellipse like) curve

$$r(t) = a\cos(t)i+b\sin(t)j+ctk$$ So this is the equation of a curve for which I've had trouble calculating it's unit vectors, especially $B$ and $N$. I know that $T = v/|v|$ and that also $B = v × a ...
Mohammad Teymuri's user avatar
10 votes
3 answers
691 views

Find center of externally tangent circle

I've been struggling to find a way to resolve the following problem: Let $C_1$ a circle of center $V$ and of radius $r_1$. Let $A$ and $B$ two points outside of $C_1$, and $L$ a line passing by them. ...
Teatoon's user avatar
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1 vote
1 answer
65 views

How do I find a constant $C$, when $x = a$, through the tangent line? [closed]

The full question is as follows: For some constant $C$, the equation of the tangent line to the graph of $y = f(x)$ = $4x^4$+C at the point where $x=a$ is $y = −78.608x − 106.1252$ Find $C$. My ...
Jonah Legg's user avatar
2 votes
3 answers
218 views

Finding Equations for Common Tangents Between Two Ellipses, A General Solution

I'm interested in finding the four common tangents between two ellipses. While I've found some fascinating approaches using dual conics to identify their intersection points (link 1, link 2), my ...
MohG's user avatar
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-1 votes
1 answer
60 views

Is it correct to say that if two parabolas touch, then tangents at the point of intersection have the same slope?

Is it correct to say that if two parabolas touch, then tangents at the point of intersection have the same slope? If it is true, is there a neat geometric interpretation of the fact?
Ayanokouji Kiyotaka's user avatar
6 votes
2 answers
263 views

Circle tangent to rotated ellipse and horizontal line

I would like to find the position for the center of a circle $(x_0, y_0)$ that is tangent to both an ellipse and a horizontal line. The ellipse is positioned at $(0,0)$ and is defined by major axis $a$...
Benjamin's user avatar
0 votes
0 answers
39 views

Find the function with derivatives

Could you help me by telling me if this is done correctly? Find a function whose tangent line at the point $(1, 0)$ is parallel to the tangent line at the point $(\pi/2, \pi/2)$ of the curve $y = \...
Quijano Garcia Estefania's user avatar
0 votes
1 answer
92 views

Comparing area of triangles

Let $f(x)=x^4+(2-a) x^3-(2 a+1) x^2+(a-2) x+2 a$ for some $a \geq 2$. Draw two tangent lines of its graph at the point $(-1,0)$ and $(1,0)$ and let $P$ be the intersection point. Denote by $T$ the ...
BlizzardWalker's user avatar
1 vote
1 answer
71 views

Why there are not 4 tangents from a point to a hyperbola or ellipse?

The equation of a tangent to a ellipse (x²/a²)+(y²/b²)=1 can be written as y=mx±√(a²m²+b²) ,where m is the slope of the tangent, now if this tangent passes through a specific point outside the ellipse,...
Aditya Mukherjee's user avatar
0 votes
1 answer
33 views

Why the tangent line has two equivalent definitions,looking for a long time can not find

Why the tangent line has two equivalent definitions, one is according to the Angle, the other is according to the limit position of the cut line, how the two definitions are equivalent, please give ...
poLir LANCER's user avatar
0 votes
0 answers
33 views

Can there be external division here?

Question: A curve $y=f(x)$ passes through $(1,1)$ and at $P(x,y),$ tangent cuts the $x$-axis and $y$-axis at $A$ and $B$ respectively such that $BP:AP=3:1$ then A) $xy'-3y=0$ B) normal at $(1,1)$ is $...
aarbee's user avatar
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0 votes
1 answer
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Point of contact between an ellipse $(5 \cos t , 3 \sin t)$ and an apparent tangent $ x \cos(R) + y \sin(R) =D$ sliding on it.

Let $E$ be an ellipse defined by $ (x^2 / 5^2) + ( y^2 / 3^2) = 1 $ or, equivalently $( 5\cos t , 3\sin t )$ with $0\leq t \leq 2 \pi$. Let $P= ( 5 \cos R , 3 \sin R) \space 0\leq R \leq 2 \pi$ be a ...
Vince Vickler's user avatar
0 votes
1 answer
88 views

Why is a derivative undefined at its discontinuities?

This question deals with why the derivative of $f$ is not defined at discontinuities in $f$. I found the answers satisfactory. My question deals with why the derivative is not defined at ...
user110391's user avatar
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0 votes
2 answers
124 views

Given points $A$, $B$, $C$ and $D$ lying on a circle and lines $BD$ and $AC$ intersecting at $F$, prove lines are parallel

The diagram shows the points $A$, $B$, $C$ and $D$ lying on a circle. $AC$ and $BD$ intersect at point $F$. $EG$ is tangent to the circle at point $C$. $AD$ is produced to meet the tangent at point $E$...
Talha Ahmed's user avatar
1 vote
1 answer
57 views

Find the equation of the line passing through the 2 points of the tangents on a circle from point $(x_1,y_1)$

Given a circle $x^2+y^2=r^2$ and a point P(x1,y1) outside of the circle. I can draw two tangents from P to the circle. I will call A and B the points where the tangents cross the circle. How can I ...
patzoul's user avatar
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0 votes
0 answers
46 views

AB is an external tangent to $S_1$ at A and to $S_2$ at B and common tangent at P cuts AB at Q.

Question: Two circles $S_1$ and $S_2$ of radius $3$ and $4$ touch each other externally at point P. If AB is an external tangent to $S_1$ at A and to $S_2$ at B and common tangent at P cuts AB at Q. ...
aarbee's user avatar
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2 votes
1 answer
31 views

Are the tangent lines at the farthest-separated points on a closed plane curve always parallel?

Suppose you have a closed differentiable plane curve. Are the tangent lines to the curve at the most distant points on the curve always parallel? What if we assume that the curve is convex? I don't ...
tparker's user avatar
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1 vote
2 answers
72 views

How to find the tangent lines of a parabola which must pass through a point without calculus. [closed]

I‘m tutoring a student who doesn’t know calculus (yet). I was given the following question: Find all lines of tangency of the graph $y=x^2$ which pass through the point $P(-6,-5)$. I know I have to ...
Mimir2902's user avatar
-1 votes
2 answers
55 views

If the length $BC=l$, length of arc $AB=l_1$ and length of arc $AC=l_2$, then $l+l_1+l_2=$

Question: A circle with centre $C_1$ and radius $\frac32$ touches another circle with centre $C_2$ and radius $\frac12$ externally at point $A$. A common tangent touches circle with centre $C_1$ at B ...
aarbee's user avatar
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2 votes
0 answers
65 views

Is this method applicable in all such geometry questions?

The Problem statement is: An ant leaves the anthill for its morning exercise. It walks 4 feet east and then makes a 160° turn to the right and walks 4 more feet. It then makes another 160° turn to ...
IndianGoldMedalist's user avatar
3 votes
2 answers
128 views

8 planes tangent 3 spheres in the space

I know it might seem like a trivial question, but I think the result is very long and I wanted a consultation to find a "smart" way to solve it without wasting hours of time on unnecessarily ...
Math Attack's user avatar

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