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.
1,329 questions
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Geometry Problem About Tangent [closed]
Let C1:(x-y)²-8(x+y-2)=0 and C2:4x²+9y²=36 be two curves on x-y plane. If A,B be two common points on C1 from where perpendicular tangents are drawn to C2.
Then, if coordinates of A(a1,b1) and B(a2,b2)...
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Tangent line of a curve [closed]
Without relying on the concept of limits or derivatives, how can I prove that a line is tangent to a curve?
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How do I find the tangent line at the pole of r=5sin(θ)? [closed]
I know that the tangent line at the pole of any polar equation must meet the parameters r=0 and dr/dθ≠0. So, r=5sin(θ)=0 at θ=0,π,2π. Also dr/dθ=5cos(θ), thus dr/dθ≠0 for any of said values. So, I ...
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Tangent to the graph of $y = x^3 +x + 16$ passes through the origin. Find the tangent of the angle of inclination of this tangent to the axis oX.
The tangent to the graph of the function $y = x^3 + x + 16$ passes through the origin. Find the tangent of the angle of inclination of this tangent to the axis of the abscissa.
As far as I understand, ...
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"How to Find the Reflection Point on a Circle for the Shortest Path Between Two Points, Satisfying the Law of Reflection?"
It is a Descriptive geometry problem, the solution of which will allow to correctly, graphically draw the reflection of sound (architectural acoustic calculation) from an unknown point on a given ...
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Given 2 concentric circles, can I find a tangent from a point on the larger circle?
I'm in the process of creating a videogame in which you manage a space station. Spaceships will fly towards the space station to dock with it if the station has what they need and an available docking ...
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Finding the tangent lines to an ellipse which pass through a point not on the graph
I am struggling to find the tangent lines to the parabola $x^2 + 9y^2 = 36$ which pass through the point $(12,2)$ not on the graph.
I've already found one tangent line, that being $y = 2$, because the ...
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Solving by elimination an analytical geometry question about circles
I came across this question on Quora which has been solved by one of the Quora user.
The thing is that he has used the method of elimination to get to the final solution rather than solving it the ...
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How does substitution for change of variables for cubics work?
I'm reading the following set of notes where they prove the following theorem
Theorem 5.1.4: The equation of an irreducible cubic with a[t] flex point [..P..?] can
be written as
$$y^2z = x^3+ ax^2z + ...
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Need help understanding this tangent line problem [closed]
The problem is, “If the line tangent to the graph of $f$ at $(1, 7)$ goes through $(2, 3)$, which of the following must be true?
A. $f’(1) = \frac{1}{4}$
B. $f’(2) = -\frac{1}{4}$
C. $f(2) = 3$
D. $f’(...
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How do I find the intervals where $h(x)$ has a horizontal tangent line from the graph of $g(x)$ and $f(x)$ (Calculus)
I included the graph to make the question easier to visualize. The question gives $h(x) = f(g(x))$. On what intervals does $h$ have a horizontal tangent line.
First, I tried using the chain rule to ...
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slope of Tangent line perpendicular to line from origin does not equal the derivative
Slope of tangent line at theta=pi/3 and r=2-3cos(theta)
I tried finding the slope of the tangent line by using a perpendicular line from the origin and then finding its slope. However, the real answer ...
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For all $h\in \mathbb{R}\sim\{0\}$, two distinct tangents be drawn from the point $(2+h,3h-1)$ to the curve $y=x^3-6x^2-a+bx$ then $\frac{a}{b}=?$
If for all $h\in \mathbb{R}\sim\{0\}$, two distinct tangents can be drawn from the points $(2+h,3h-1)$ to the curve
$y=x^3-6x^2-a+bx$
then find value of $\frac{a}{b}$
My Attempt
If two distinct ...
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To prove Tangent to the circle and Radius are Orthogonal
Prove that without using Proof by Contradiction, Tangent to the Circle and Radius are Perpendicular
Here is my try:
Consider a circle with Centre $O$ and a Tangent drawn to it at the point $P$ shown ...
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Angle between a tangent line and another line
I was solving exercises from Do Carmo’s and one exercise goes: Show that the tangent lines to the regular paramterized curve $\alpha(t)=(3t,2t^2,2t^3)$ make a constant angle with the line $y=0$, $z=x$....
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Curves meeting at a tangent and repeated roots
Hey I came across a question where there was two curves, one defined parametrically:
$C_1$: $y=c/t$
, $x=ct$
$C_2$: $(x-\alpha)^2 +(y-\beta)^2=r^2$
One part of the question relied on the fact that ...
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given a point and a circle, find the point(s) on that circle which would have a tangent that intersects the point
I'm trying to find the points on a circle of radius $o_{r}$ that have a tangent line which intersects the point $P_{s}$
BACKGROUND:
This is my first post, if i messed something up let me know!
I'm ...
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(How) can I determine the point of tangency of $k\cdot x$ and $\sin{x}$?
Is it possible to determine the exact coordinates of the point of tangency of a line through the origin $k\cdot x$ and $\sin{x}$, with a $k$ chosen so there exists a unique point of tangency? The ...
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Line tangent to a parabola [closed]
So I was doing some AoPS Alcumus and came across this problem with a weird solution. A quadratic function $p(x)$ has lines of tangency $y=-11x-37$, $y=x-1$, and $y=9x+3$. These lines are tangent to $p$...
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Prove that the tangents from point $A$ and the tangent from point $B$ to the circle $k_1$ are parallel.
Given is a circle $k$ with diameter $AB$. On it, we choose a point $M$, which is not coincident with $A$ or $B$. Let $k_1$ be the circle that has its center at $M$ and is tangent to the diameter $AB$. ...
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Examining the function.
Given function: $$f(x)=|x+2|e^{-\frac{1}{|x|}}$$
Find constants $a, b$ and $c$ such that $f(x)= ax +b+\frac{c}{x}+ o(\frac 1x)$ when $x \rightarrow +\infty$ and $x \rightarrow -\infty$?
Examine the ...
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Question on tangent planes and normal lines
PFA the question I am looking at. For this one, I computed that the point P is $(1,1,1)$. Then I find the gradient to the surface and compute it at $(1,1,1)$ which gives me the normal vector to the ...
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Tangent lines equal secant line
Consider a continuous function $f$ with exactly three distinct points
$$
x_1(m) < x_2(m) < x_3(m)
$$
with slope $m$, meaning
$$
\frac{f(x_3(m))-f(x_1(m))}{x_3(m)-x_1(m)}=f'(x_3(m)) = f'(x_2(m)) =...
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To derive the equation of the director circle of a standard ellipse using the tangent lines at any two points on the ellipse
I am currently studying conic sections and have encountered the concept of the director circle (or orthoptic circle) of an ellipse"1". The director circle is defined as the locus of points ...
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What is the explanation of these special cases of tangents [closed]
The graph $y = |x|$ has no tangents at $x=0$ since it is not differentiable at that point, but it is possible to draw an infinite number of tangents to the graph at $x=0$, so why does it technically ...
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There is no polynomial p(x) for which there is a single line that is tangent to the graph of p(x) at exactly 100 points.
False. For p(x) with exactly 100 multiple roots the X-axis is such a line. (This is
essentially the only way: if y = ax + b is such a line for a polynomial q(x), then the X-axis
is such a line for the ...
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Tangent definition with a newton raphson question example
I'm very confused about the definition of a tangent, I was told its a straight line that touches a curve at a point, but if extended does not cross the curve at any other point.
In this question if u ...
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Significance of tangent passing through the origin in graphical interpretation
(This question was asked in CSIR NET June 2019 examination)
A monkey climbs a tree to eat fruits. The amount of energy spent climbing on the different branches has a relationship, as shown in the ...
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Lines AM and AN are both the tangent of a circle with the center O, in what situation is OA the bisector of MN?
I've been recently stuck with the corresponding question in the title, after working it out for a while I came up with an answer: You can't.
Since AM and AN may be of any length.
But my question is, ...
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How many tangents to a curve are there that are parallel to a secant line that passes through two, infinitely close, points on the curve?
In order to find the slope of the tangent to, for example, $f(x) = x^2$ at the point $x = a$, I first look for the slope of the secant line that passes through two infinitely close points $Po(a, f(a))$...
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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 ...
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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, ...
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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 ...
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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}{...
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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 $N$, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $\gamma_2$ ...
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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 $...
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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 ...
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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 ...
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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) ...
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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 ...
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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 ...
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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^\...
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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 ...
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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 ...
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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 ...
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