# 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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### 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 $y$...
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### 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 ...
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### 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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### 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
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### 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 ...
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### 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
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### 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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### 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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### 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
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### 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 ...
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### 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 ...
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### 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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### 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
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### 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),...
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### 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 ...
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