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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3 votes
1 answer
90 views

What's wrong in this line of reasoning (involving tangent lines to $x^2$)?

Context. I have been trying to solve the problem described below. I found a similar question and I see where I have gone wrong. I assumed the points on the parabola would be symmetric across the $y$...
1 vote
1 answer
64 views

Proving that the feet of the perpendiculars from the foci on any tangent lie on the auxiliary circle using parametric coordinates

In the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, the parametric equation for a tangent is $\frac{x}{a}\cos(h)+\frac{y}{b}\sin(h)=1$, and the foci are $(ae,0)$ and $(-ae,0)$, taking the focus on the ...
0 votes
0 answers
35 views

Approximation of plane curve with tangent vector

Let $c:[-1,1]\to\mathbb{R}^2$ a $\mathcal{C}^1$ planar curve and suppose that $c(0)=(0,0)$ and $c'(0,0)=(a,0)$, $a>0$. I'm trying to prove the following statement (without any success): there exist ...
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2 votes
2 answers
68 views

What points in the plane of the graph $y=x^3$ have three tangents to the curve passing through them?

I’m studying high school math and encountered this question in the extension section for derivatives. The text says an algebraic solution to the problem is harder but possible. There is also a similar ...
1 vote
1 answer
53 views

The gradient at the meeting point

When two straight lines touch at one point, their gradients are most definitely not the same. However, when I draw a tangent to a curve, why is the gradient of the tangent the same as the gradient at ...
-1 votes
4 answers
57 views

If the equation $7x + y = k$ is the equation of a line tangent to the graph of $y = 9x + 1/x^2$, what is the value of $k$ [closed]

How would I solve this? I tried everything, but I am still lost. Please help. I watched some videos, but I still don't know how to work it with this: If the equation $7x + y = k$ is the equation of a ...
1 vote
1 answer
109 views

The relation between formal group $\frac{X+Y}{1-XY}$ and algebraic group $x^2+y^2=1$ with group law $*$

$T(X,Y)=\frac{X+Y}{1-XY}$ is a power series which satisfies axiom of formal group. My book reads this formal group comes from algebraic group $S: x^2+y^2=1$, with group law $*$:$(x_1,y_1)*(x_2,y_2)=(...
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1 vote
0 answers
48 views

Chord of a conic touches another conic

If $P, Q, R$ are three points on the conic $\frac{l}{r}=1+e\cos\theta$ and the tangent at $Q$ meets $SP$ and $SR$ in $M$ and $N$ so that $SM=SN=l$, where $S$ is the focus, then prove that the chord $...
0 votes
0 answers
14 views

continuous differentiable, homogene, equation

Hello I have two questions: Let $f : \mathbb{R}^n \setminus {0} \rightarrow \mathbb{R} $ be continuous differentiable and homogene of degree $k$. Prove that we can extend $f$ to a continous ...
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0 votes
2 answers
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Find the equation of the tangent line(s) to the curve $f(x) = \sqrt{x + 5}$, that passes through the point $(-8, 1$). [$-8$ can't get subbed in]. [closed]

Going through the process of finding the derivative, which is $\dfrac{1}{2\sqrt{x+5}}$ ,and substituting x gives an undefined slope, as you end up taking the square root of a negative number I know ...
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0 votes
1 answer
57 views

Tangent to two parametric curves

If I have two parametric curves one defined as x(t) and y(t) and another x(s)+c and y(s). My assumption is that if I set t = s = value. I can find the two slopes at these values. i.e. dy/dx = x'(t)y'(...
1 vote
1 answer
44 views

Why is the distance equal to the height between these two circles?

I have a follow up question for a previous question linked here:What is the equation for the distance between these two circles? In that question asked for an equation to find the distance between the ...
2 votes
1 answer
65 views

Can torsion be negative?

If $B$ denotes the binormal vector, $S$ denotes the arc length, $\tau$ denotes torsion, and $N$ denotes principal normal vector, then, $$\frac{dB}{dS} = -\Bigg|\frac{dB}{dS}\Bigg|N = -\tau N$$ So if $\...
2 votes
3 answers
121 views

How can you find the equation of the line that is tangent at two distinct points to the curve? [closed]

Given a curve $y = x^3-x^4$, how can I find the equation of the line in the form $y=mx+b$ that is tangent to only two distinct points on the curve? The problem given is part of the Madas Special Paper ...
0 votes
0 answers
21 views

Regular differentiable curve has strong tangent

I have a doubt about exercise 1.3.7 from the book Differential Geometry of Curves and Surfaces by Do Carmo. The exercise asks to prove that a curve $\alpha$ which is $C^1$ at $t_0$ and $\alpha'(t_0)\...
4 votes
1 answer
95 views

How do I find the function and derivative of an unknown curve?

I have $x$ and $y$ values to plot the curve and I need to find a tangent line of slope 1 that intersects the curve (and the point at which it intersects). I was trying to do polynomial and exponential ...
7 votes
1 answer
148 views

A chain of circles of radius $1/n^p$ is tangent to the $x$-axis. What is the horizontal length of the chain?

I recently discovered that, if a chain of circles of radius $1/n^2$, where $n\in\mathbb{N}$, is tangent to the $x$-axis, then the the horizontal length of the chain is exactly $2$. This can be shown ...
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1 vote
3 answers
84 views

Find the equation of the circle touching the line $(x-2)\cos\theta+(y-2)\sin\theta=1$ for all values of $\theta$

Find the equation of the circle touching the line $(x-2)\cos\theta+(y-2)\sin\theta=1$ for all values of $\theta$ The answer is given on toppr website. It says, $(x-2)\cos\theta+(y-2)\sin\theta=\cos^2\...
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0 votes
1 answer
63 views

Is there a general way to prove that PX, PY, PZ are in GP without specifically letting A to be on y-axis?

Let BC be the chord of contact of the tangents from a point A to the circle $x^2+y^2=1.$ P is any point on the arc BC. Let PX, PY, PZ be the lengths of perpendiculars from P on the AB, BC and CA ...
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1 vote
1 answer
42 views

tangent slope of cardioid graph

Hi i am student working on calculus and i have got question that i came up with wrong answer. So we get $r=2(1+\cos\theta)$ cardioid function and the question is to looking for Θ angle where tangent ...
1 vote
1 answer
91 views

Vertical Tangents and Derivatives

According to wikipedia (https://en.wikipedia.org/wiki/Vertical_tangent): A function ƒ has a vertical tangent at x = a if the difference quotient used to define the derivative has infinite limit: ${\...
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1 vote
1 answer
40 views

Proof of concurrency of transverse tangents and line joining centers of two disjoint circles

I read that "the transverse common tangents to two disjoint circles and the line joining the center of the circles are concurrent" and tried proving it. Can I get a hint to prove that lines ...
1 vote
1 answer
78 views

Tangent cone of an arbitrary algebraic curve

So, my problem is: given a real/complex (I will assume complex) algebraic curve, say $f(x,y)$, or $f(z,w)$ for $x,y\in\mathbb{R}$ or $\in\mathbb{C}$ (I would like to hear your thoughts in either case),...
1 vote
2 answers
19 views

Calculating tangent at two points $(4a,8a)$

Find the equation of the tangent to the curve $ay^2=x^3$ at the points $(4a,8a)$ I have re-arranged the equation to get $$y = \left(\frac{x^3}{a}\right)^{\frac{1}{2}}$$ Then taking its derivative I ...
0 votes
0 answers
13 views

Calculating the tangent at the point x = $\theta$

I am finding the tangent at the point $\theta$ when $x = a\cos(\theta), y = b\sin(\theta)$ The tangent is calculated with the following $$\frac{y-f(\theta)}{x-\theta}=f'(\theta)$$ at a point where $x=\...
1 vote
0 answers
38 views

Calculating tangent at point x = a

The equation of the tangent to a curve $y = f(x)$ at a point where $x = a$ and $y = f(x)$ at which the derivative $f'(x)$ exists and is finite, is, $$\frac{y-f(a)}{x-a}=f'(a)$$ Find the equation to ...
-1 votes
1 answer
45 views

Trouble understanding the tangent plane of a surface

$$z = z_0 +a(x-x_o)+b(y-y_0)$$ where $a$ and $b$ are the partial derivates with respect to $x$ and $y$. I dont understad why $a$ and $b$ are there since they are tangent to the plane not normal?
1 vote
1 answer
55 views

How to find radius of a circle drawn inside a triangle, if we know the base and the distance from point to the circle

How to find radius of a circle drawn inside a triangle, if we know the base and the distance from point to the circle Here we can see the problem described in the title, we know only the distances a=...
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3 votes
1 answer
86 views

Tangent to a graph through an exterior point

Say you are given $f(x) = 5x^2$ and you want to find a tangent line to $f$ that goes through $P(0|-10)$. The two options you have is using the tangent line equation, $t(x) = f'(a) \cdot (x - a) + f(a)$...
0 votes
0 answers
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What is meant by dividing the abscissa of the point of contact?

The tangent at any point on the curve $x=at^3, y=at^4$ divides the abscissa of the point of contact in the ratio $m:n,$ then find $|m+n|$ ($m,n$ are coprime) I don't know what the meaning of '...
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1 vote
0 answers
39 views

Find equation of the line tangent to the level curve of $f(x, y)$ at given point

I was given the following problem: Let $f(x, y) = \frac{x-y}{x+y}$ and $P=(1, 1, f(1, 1))$. Find the gradient of $f$ at $P$, the equation of the plane tangent to $f$ at $P$, and the equation of the ...
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0 votes
0 answers
110 views

Condition for two common tangents

Let $C_1$ and $C_2$ be the centres of two circles, $r_1$ and $r_2$ be their radii respectively. I know that the condition for the existence of four common tangents between the two circles is $r_1 + ...
0 votes
1 answer
40 views

Find coordinates of a point for a derivative of a parametric curve

Find the coordinates of the points at which the given parametric curve has a) a horizontal tangent and b) a vertical tangent. The parametric curve is $$\mathscr{C}=\begin{cases}x=t^2+1\\y=2t-4\end{...
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1 vote
0 answers
38 views

Proving Tangent line perpendicular to the radius using Contradiction

Here is the proof by contradiction to prove tangent to the circle and radius are Perpendicular at the point of Contact. Let me know whether the proof is Complete. Consider the circle with center $O$ ...
1 vote
1 answer
79 views

discovering Mean Value Theorem

mean value theorem for single variable function is very easy and intuitive once you "see" the formula. Actually, My question, slightly weird but helpful, is that How does someone come up ...
0 votes
0 answers
28 views

Finding tangent point of a line with a given function

Given the following function: $f(x)=x\left(1-\frac{n}{x}\right)^d$ with $n\geq1$, $x>n$, $d>>1$. I need to find the tangent point with the line $ax+b$, with $a$ given. Note that $0<a<1$....
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0 votes
1 answer
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Is it correct to say the differential is the equation of the tangent line?

For example. Given function $f(x)=\sqrt[3]{x} \Rightarrow f'(x) = \frac{dy}{dx}= \frac{1}{3}x^{-2/3}$ $dy = (\frac{1}{3}x^{-2/3})dx$ compared to... $\Delta y = \frac{1}{3}x^{-2/3}\Delta x$ I know ...
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-1 votes
1 answer
245 views

Determine the numbers a, b and c such that it satisfies the condition

I have a function $f(x) = x^3 + ax^2 + bx+ c$ and I need to solve for the numbers $a$, $b$, and $c$. The numbers need to satisfy the following condition: The slope of the secant line defined by points ...
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2 votes
2 answers
60 views

Why am I getting $\beta=90^{\circ}$

Consider the geometry below, where the small circle is touching both semi circles of radius $5$ and the side of the square. Find the radius of the small circle. My try: $M$ and $N$ are centers of ...
0 votes
1 answer
116 views

Finding the tangent line of curve $x^2y^2+5xy=14$ at $(2,1)$

I want to check if my work is correct. Find the equation of the tangent line to the curve at (2, 1)$$𝑥^2𝑦^2 + 5𝑥𝑦 = 14(1) $$ solution: The tangent is a straight line so it will be of the form: $...
1 vote
0 answers
63 views

How does Fermat's method of adequality actually work?

I've been looking at Fermat's method of obtaining a tangent line through this resource https://cedar.wwu.edu/cgi/viewcontent.cgi?article=1012&context=wwu_honors and it says to basically take an $...
1 vote
1 answer
24 views

Perimeter of triangle $SPR$

I have to calculate perimeter of triangle $SPR$. I know, that length between $A$ and $P$ is $25$. I know that length between $B$ and $P$ is also $25$. Also in $A$ and in $B$ is angle $90$. Any help?
0 votes
0 answers
66 views

Let $r(t)$ be a parameterized curve with $r(t) = (x(t), y(t), z(t))$ and $x = (1 + \cos(t)), y = \sin(t), z = 2 \sin(t/2)$

Let $r(t)$ be a parameterized curve with $r(t) = (x(t), y(t), z(t))$ and $x = (1 + \cos(t)), y = \sin(t), z = 2 \sin(t/2)$. Determine the unit tangent vector and curvature at a general point. I am ...
0 votes
0 answers
37 views

Drawing tangents to curves by hand

Recently I was doing a practice test that asked to draw a tangent on the graph at x = -1.5 and calculate it's gradient. However when drawing the tangent I found the gradient to be extremely inaccurate ...
3 votes
1 answer
86 views

an example of tangent equation calculation, from a book, that I do not understand

I am in trouble with an example of an equation of a tangent from a book. Here's what my book is writing (in french) : I translate it (summarizing a bit) : take a T(X,Y) point on the tangent, the ...
0 votes
1 answer
26 views

Condition for tangency in 3D space

So I have a cone of the form $x^2+y^2 = k^2 z^2$ for some $k\in\mathbb{R}$ and some ellipse $\vec{x}(\theta)= \vec{c_0} + (\cos\theta)\, \vec{u} + (\sin\theta)\, \vec{v}$ parameterised by $\theta$, ...
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1 vote
0 answers
14 views

The class of curves whose derivative can be computed solely with analytic geometry

Reading old math papers and refreshing my geometry has lead me to the following: Question: What is the largest class of curves $f:[a,b]\to \mathbb{R}$ whose tangent line at a point $x_0\in [a,b]$ can ...
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0 votes
0 answers
15 views

Finding a general expression for the intersection of tangent planes to a pair of intersecting n-D hypersurfaces.

If $f=0$ and $g=0$ be 2 surfaces in $\mathbb{R}^3$ and their curve of intersection passes through the point $p$, then the tangent planes at $p$ are : $$ \sum_{1≤i≤3} (x_i-x_{ip}) \partial_{x_i}f(p)=0 $...
0 votes
1 answer
47 views

Tring to find out what this relationship is called, so I can figure out how it works.

Picture of what I am looking for I have a formula where if I have a circle sitting in an saddle angled at 10 degrees and I know the distance from the bottom of the saddle to the top of the circle it ...
0 votes
1 answer
62 views

Differential as a tangent to $\Delta f(x)$

I had come across the following The number $h = x − x_0$ , that is, the increment of the argument, can be regarded as a vector attached to the point $x_0$ and defining the transition from $x_0$ to $x ...
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