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Questions tagged [tangent-line]

For questions on the tangent line, the unique straight line that touches a function locally only once.

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Find the equation of the circle between 2 tangent lines

Find the radius of the circle and its position from origin. Given - equations of tangents of the circle and point of intersection of the tangents.(It's like a pair of tangents from a circle ...
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If the tangents are parallel at each point for two curves, then so do their principal normal and binormal vectors

In the book of Differential Geometry by Kreyszig, at page 103, it is asked that Problem 13.1: Given two twisted curves which are in a one-to-one correspondence so that at corresponding points ...
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Geometrical problem on semi-circles

Given that $AE$ is the tangent to the small semi-circle at $D$ and that arc $CD$ : arc $DB$ = $3 : 10$, find arc $AE$ : arc $EB$. How do I go about solving this? I do not know how to start.
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Problem on tangents drawn to a circle

I am solving Co-ordinate geometry by S.L. Loney. I am stuck on a problem on circles involving tangents and chords. I am not sure, if my approach is correct to solving this problem. Any inputs, tips ...
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Area of a region delimited by chords and circular arcs.

Let $AB$ be the diameter of circle $O$, where $AB = 2$. Circle $P$ is internally tangent to circle $O$ at point $B$, and $PB$ = $\frac{2}{3}$. Two different chords $AX$ and $AY$ are drawn tangent to ...
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$A$ and $D$ are in circumference of a circle and $B$ and $C$ are its inner points such that $PA$= $12$, $\frac{AB}{CD}$ = $\frac{1}{2}$. Find $PC$

There is something misunderstanding with that question that I think it to have inadequte context or information (obviously for my little knowledge). So I couldn't solve the problem. SOURCE: ...
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Maximum number of tangents to two circles in affine geometry

How would one prove that the maximum number of tangents to two circles is 4, without recurring to the equations of the circles? I have found several ways of determining them (most of them using ...
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Quadric surface S tangent to plane

Suppose that the quadric surface S is given by $z = x^2 + x + 2y^2 + 3y$ and the plane is given by $x + y + z = k$, where k is a constant. Find the vector equation for the tangent line to the curve ...
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Tangent line help(without calculus)

I need to find to find a tangent line to the curve $x \over {x^2 + x + 2}$.
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Tangent Line From a Point on a Sphere and $y$-axis

Let's say I have a sphere, $$100 = x^2+y^2 +z^2 $$ This indicates that the center of our sphere is at $$(0, 0, 0)$$ and we have a radius of $$radius = 10$$ I'm under the assumption that $$P = (1, ...
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When to use different formulas to find the slope of a tangent line

I'm having some difficulty understanding the formulas to find the slope of a tangent line. As per my textbook, the first formula we received is presented below: The tangent line to the curve $y = ...
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Finding the value of constant a [closed]

The point P lies on the curve f(x). at point P the graph has gradient fiven by 7ab. The gradient of the normal at this point is given by 2/b. Determine the value of constant A. I cannot find any way ...
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Two circles with a common tangent

Find the angle $\angle BAC$ in the following picture . My attempt : I tried to apply the relationships in the both circles between different angles and arcs in many ways but it didn't work . Also ...
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Finding slope m of tangent to curve

been years since I took calculus and am currently struggling with how to properly work out the following: Using the tanget line slope formula: My understanding is that I would need to find the ...
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You are given the polar curve r=cos(2θ). Find the points where the tangent line is horizontal and where the tangent line is vertical.

Some answers are listed below that I have gotten right. Unfortunately I am not getting the right answers for the majority of them a. (a) List all of the points $(r,\theta)$ where the tangent line is ...
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Calc2 Finding $f'(2)$ from tangent line

So, I have a Calc 2 problem I am stuck on. The tangent line to $h(x)$ at $x = 2$ is $3x - 2$. It says to find $f'(2)$ given $f(x) = -3[h(x)]^2 + 2x + 2$ Any ideas on how to go about this? Thank you!
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Use Limits to calculate slope of the tangent

Use limits to calculate the slope of the tangent to the curve $y=\frac1x$ at $x=a$. I need to write an equation for the tangent to $y=\frac1x$ at $x=4$. I think I understand the basics of the ...
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Finding where two graphs have perpendicular tangent lines

I've been stuck on this calc problem for a while: Let $f$ be the function given by $f(x) =\ln(x+1)$ and let $g$ be the function given by $g(x) = x^{-1/2}$. At what value of $x$ do the graphs of $f$ ...
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Prove: The normal to the involute of a circle is tangent to the circle

Please refrain from using algebraic equations (in the Cartesian system) to prove it. I was looking for some kind of geometric proof for it. Or using differentitiation of vectors. For using the ...
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Tangent line, small o

Let $f,g,h \in C^2[a,b], ||f|| = |f(x_0)|$ and let $$ f'(x_0) = 0, ~~ f(x_0)f''(x_0)<0, ~~ g(x_0) = 0, ~~ g'(x_0)\neq 0.$$ I have to find local max $$ ||\phi_\lambda|| = ||f +\lambda g + \frac{\...
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f(x) whose tangent line has the slope and graph passes through a point.

I'm trying to find the function $f(x)$ whose tangent line has the slope $\frac{(1+\sqrt x)^{1/2}}{8\sqrt x}$ for any $x\neq 0$ and whose graph passes through the point $(9,\frac{3}{4})$. I'm not ...
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Tangent line, normal line - confusing.

I am practicing finding tangent and normal line. The tangent/normal line is usually to some graph, and parallel/perpendicular to some other line at the same time. Not that complicated. Can someone, ...
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Tangents to parabola $y^{2}=4ax$ meet hyperbola $x^2/a^2-y^2/b^2=1$ at $A$ and $B$. Find the locus of intersections of the tangents at $A$ and $B$.

If tangents to the parabola $y^{2} = 4ax$ intersect the hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ at $A$ and $B$, then find the locus of point of intersection of tangents at $A$ and $B$. I ...
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Tangent vector to two circles in 3D

Given the coordinates of two sheaves $\overrightarrow{p_a}$ and $\overrightarrow{p_b}$ and the axis of rotation of their rotation $\overrightarrow{s_a}$(blue) and $\overrightarrow{s_b}$(cyan) ...
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Why we don't get two tangents in the point form of tangent from a given point to a circle?

In my textbook I have read the point form of representation of tangent from a point $P(x_1,y_1)$ to a circle $x^2 + y^2 + 2gx + 2fy + c = 0$ which is given by $$xx_1 + yy_1 + g(x+x_1) + f(y+y_1)+c=0$$...
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To find the tangent of a given curve.

The given curve is $$y^2(a+x)=x^2(3a-x).$$ It is given in my book that when we equate the minimum order term of equation to zero, $$a(y^2-3x^2)=0,$$ we get $y=\sqrt{3}x$, $y=-\sqrt{3}x$, which is ...
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Finding arrays of pixels perpendicular to a derivative

I have a question that I originally posted to Stackoverflow, but got no answers, and was encouraged to re-post here. I am developing a method for “straightening” binary-masks of pictures of carrot ...
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The range of values of $a$ such that…

Question The range of values of 'a' for which the common tangent to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2}=1$ and the parabola $y^2=4x$ and their chord of contact can form an equilateral ...
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Ceva's Theorem: Proving lines in a specifically constructed triangle intersect

Question: Let $o_1$, $o_2$, and $o_3$ be circles with disjoint interiors with centres $O_1$, $O_2$, and $O_3$, respectively. Among the lines tangent to both of the circles $o_2$ and $o_3$ there are ...
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Equation of perpendicular line to tangent lines

I'm completely stuck on a question about equation of perpendicular lines to tangent lines. I figured out my tangent lines equations, I know graphically what I should get for my perpendicular lines (x=-...
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Find Equations of tangent lines

I'm having a hard time figuring this out. I'm asked to find the the equations of the horizontal lines to the curve of $$y=x^3-3x+1$$ I set the derivative equal to zero and solve for x, to find the ...
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Bidirectionally of the “Tangent Criterion”

I've recently been reviewing some basic geometry concepts when I saw this one in Evan Chen's fantastic "Euclidean Geometry in Mathematical Olympiads" (EGMO). Proving $(i)\Rightarrow (iii)$ is quite ...
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velocity and acceleration in a parabola

A particle moves with constant speed along a parabola of equation $y^{2}=2px$ with $p=constant$. I want to find its velocity and acceleration vector. Since the velocity magnitude is constant I know ...
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Find the equation of the normal line to the curve $y = \sqrt{1+4x}$ at $x=2$.

Find the equation of the normal line to the curve $y = \sqrt{1+4x}$ at $x=2$. So, to begin we get rid of the square root right? $y=(1+4x)^{-1/2}$ Then, the power rule? $y’= -1/2(1+4x)$ ...
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Sides of orthic triangle are parallel to tangent through opposite vertex

Let $ABC$ be a triangle with altitudes $AD$, $BE$, $CF$. Prove that $EF$ is parallel to the tangent to the circumcircle of $ABC$ at $A$. (And similarly $DF$ and $DE$ are parallel to the tangents ...
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find a circle given two tangent points and angles

I've searched and found similar questions, but with less information given, and I was hoping that the additional information would allow for a more streamlined solution. Given two points, and their ...
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Find the equation of the Tangent Line to the given set of Parametric Equations at given point. [closed]

I'm looking for validation for my answer to this question. Parametric Equations: $x = t^2 + 2t + 1 , y = t^3 + 7t^2 + 8t, t = -1$ For this problem I used the Point-Slope-Form formula. myAnswer:$ y =...
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Finding the equation of the line tangent to $x^2+y^2-6x+4y-3=0$ passing through $(7,4)$. How to proceed, not knowing point of tangency?

Find the equation of the tangent line to the circle $x^2+y^2-6x+4y-3=0$ which passes through the point $(7,4)$. Graphing the circle, $(7,4)$ is a point not on the circle. So, I am assuming it's on ...
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Relation between tangent angle with horizontal and point coordinates

I would like to find the angle $\theta$ that the tangent to a curve $f(x)$ at a given point $(x,f(x))$ makes with the horizontal in terms of the coordinates $(x,f(x))$ of the point. See figure Let $(...
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Other Points Where Tangent Line Intersects Graph

Q: For each a ∈ $R$ find any other points at which the tangent line ($y = 3a^2 -48$) intersects the original graph ($x^3 - 48x + 2$). Hint: $f(x) − f(a)$ is divisible by $x − a$ Does this just mean ...
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Having trouble helping my daughter solve the calculus problem that requires finding tangent points between a cubic function and a line

determine any and all locations (x-values) where the cubic function $y=x^3-5x^2 +x$ has tangent lines that are parallel to the line $y = -2x + 8$
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What is the equation of the tangent drawn to the function $f(x)$ at the point $ x = 1 $

Here is question and my attempts: I can not get the correct answer. Let $$f(x)=\int_x^{x^2}\frac{2t^2+1}{t^3+2}dt$$ What is the equation of the tangent drawn to the function $f(x)$ at the point ...
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Find the tangent line to a curve

Find the tangent line to the curve $x^2y - y^2 + x = 11$ at the point $(3,1)$ I tried to solve it using parametric equations \begin{cases} y = t \\[4px] x = -\dfrac{1}{2t} + \dfrac{\sqrt{1+4t^3 + ...
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Tangential planes of $f(x,y) := (y^2-x)(y^2-2x) $ in $(-1,1)$ and $(-1,-1)$

Let $f:\mathbb{R^2} \to \mathbb{R}$ with $f(x,y) := (y^2-x)(y^2-2x) $ How can I find the function rules $\tau_{(-1,1)}(x,y)$ and $\tau_{(-1,-1)}(x,y)$ of the tangential planes on the graph of the ...
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Astroid slope as a function of x, y

I'm trying to find the slope of an astroid. https://en.wikipedia.org/wiki/Astroid Using $x = a\sin^3(t)$ and $y = a\cos^3(t)$, $dy(t)/dx(t) = -\tan(t)$. I expect $\tan(t) = y/x$, giving $dy/dx (x, y) ...
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Equations of lines tangent to $x^2+2x+2y^2−4y=5$ that are normal to $y=x+12$

I need to verify if what I am doing is correct here. This is my question: Write down the equations of tangent lines to the curve of the implicit function $x^2+2x+2y^2−4y=5$ that are normal to the ...
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Normal Component of an acceleration vector

A parametric equation is defined by $r\left(t\right)=cos\left(-7t\right)i+sin\left(-7t\right)j+6tk$. Compute the normal component of the acceleration vector. So I got that $r'(t)$ was $\left(-7sin\...
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Coordinates on a parametric curve

A curve is defined by the parametric equation $(x, y, z)$ $=$ $(-2 + 3t, 1 + 3t^2, 2t-3t^3)$. There is a unique point $P$ on the curve with the property that the tangent line at $P$ passes through the ...
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Equation of two parallel tangent lines and coordinates

I'm having a super hard time with this question. I've spent hours trying to solve it and I can't figure it out. The question is asking me to find the equations of two parallel tangent lines, and ...
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1answer
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Finding point of intersection where k is a non zero constant

I am struggling to solve this question: The curve $C$ has the equation $$ k x^2 - xy + (k+1)x=1. $$ The line $l$ has the equation $$ -(k/2)x + y = 1. $$ Here $k$ is a non-zero constant such that ...