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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Proof regarding a parabola, triangle with the orthocenter on the directrix and its circumcircle passing through the focus

Prove the following: The intersection points of any three tangents of a parabola given by the formula $y(y-y_0)=2p(x-x_0)$ are vertices of a triangle whose orthocenter belongs to the directrix ...
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Intersecting Secants Theorem

Let the point $A$ lie on the exterior of the circle $k(R).$ From $A$ are drawn the tangents $AB$ and $AC$ to $k.$ The triangle $ABC$ is еquilateral. Find the side of $\triangle ABC$. I am not sure ...
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26 views

Using implicit differentiation to find equation of tangent at arbitrary (a,b)

For the following equation $\sqrt x + \sqrt y = 2$ (1) Find equation of tangent at point (a, b) on curve Using implicit differentiation: $$y' = - \frac{√y}{√x}$$ Equation at (a, b) is: $$y - b = - ...
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The angle between common tangent and two chords, which are connecting tangent points and one common point of two circles

enter image description here So I saw the solving way and it says that the angle BAC equals 90 degrees, and DA=DB=DC, but can't understand why. Can someone help?
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Angle between tangent line of a curve to an axis

I'm trying to find the angle between a tangent to a curve and an axis. I have parameterised an intersection between two pipes with radius $r_1$ and $r_2$ in the following manner. \begin{bmatrix} ...
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51 views

Geometry and tangent-chord theorem problem?

According to the figure, CA is tangent to the circle, centre O, at A. ABT and POT are straight lines. Question: Given that BT is equal to the radius of the circle, prove that: $\angle ABP = 3 \...
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Derivatives and definition

I’m currently doing a course in Mathematical Analysis at University level. I ask myself a simple question; when you’re finding the derivative of a function, you’re essentially finding the rate at ...
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4answers
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I need help finding a tangent line to a parabola.

This is the question. A parabola $y=ax^2 + bx + x$ has vertex $A(2,1)$ and passes through $B(1,0)$. Find the equation of lines passing through $(0,4)$ that are tangent to the parabola. Using a ...
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1answer
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Is it possible to define a projective line that intersects a projective cubic only once?

I have been working on this and by Bezout I believe that if the line is to intersect a cubic in only one point it must do so with multiplicity 3. I am though unsure given a point $P$ on a cubic how I ...
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Right tangential trapezoid

In a right tangential trapezoid $ABCD$ $(AB\parallel CD)$ and $AD\perp AB$ the incircle is $k(O).$ Find the area of the trapezoid if $CO=6$ and $BO=8.$ The triangle $BOC$ is a right triangle and by ...
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53 views

Tangent to the circle concentric with the circle, $2x^2+2y^2-6x-10y=183$

Let $C$ be the circle concentric with the circle, $2x^2+2y^2-6x-10y=183$ and having area $(1/10)th$ of the area of this circle. Then a tangent to $C$, parallel to the line, $3x+y=0$ makes an intercept ...
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Find the equation of all circles tangential to the lines $y = 0, x = 0$ and $y = - x + 2$

I have a question, to find the equation of all circles tangential to the lines $y=0,\,x=0$ and $y=-x+2$. There should be $4$ circles. I understand so far that circles take the form $$(x - h)^2+(y - k)...
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Show that a tangent of the graph of $f$, that passes through $A$, exists.

Let $f;[a,b]\rightarrow \mathbb{R}$ be continuous on $[a,b]$ and differentiable n $(a,b)$. We consider the points $A(a,f(a))$ and $B(b,f(b))$. There $c\in (a,b)$ such that the point $M(c,f(c))$ ...
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1answer
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Why is the derivative of tangent vector always along $y$ axis?

Imagine any curve $y=f(x)$ in a cartesian coordinate system. At any point, A vector along the tangent can be given as $$ \vec V = \hat i + \frac{dy} {dx} \hat j $$ I'm trying to find the direction of ...
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2answers
70 views

How to solve $\frac{dy}{dx} = \frac{1}{\sqrt{2^2 - x^2}}$?

I've just started with differential equations and in the textbook I was given two, with one of which I have trouble. The task was to solve them with software, but I considered it'd be better to solve ...
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1answer
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For curve 𝑦=2x^2−5x+2 , if the normal to the curve at point 𝑃 is parallel to the tangent to the curve when 𝑥=2 , find the co-ordinates of 𝑃

I'm struggling on the above question. I have dy/dx as 4x-5 gradientTangent = 3 gradient normal = -1/3 the corresponding y co-ordinate at x=2 is y=0 I have also created the eqn of the tangent and ...
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How to find the arc centre and radius given the arc start point and arc end point and arc direction?

I know the arc start point and the end point. It is separated by a height/distance as shown in the figure.The blue line end points are the arc start and end points respectively. How can i drawn a arc ...
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The tangent line to the curve of intersection of the surface $x^2+y^2=z$ and the plane $x+z=3$ at the point $(1,1,2)$ passes through

The tangent line to the curve of intersection of the surface $x^2+y^2=z$ and the plane $x+z=3$ at the point $(1,1,2)$ passes through (A)$(-1,-2,4)$ (B)$(-1,4,4)$ (C)$(3,4,4)$ (D)$(-1,4,0)$ I can ...
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showing tangent line on differentiable funcions

]1 Can someone solve this? Explain your answer in detail please.
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1answer
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If a circle cuts an ellipse (both centered at the origin) at four distinct points then find the maximum value of the acute angle formed.

QUESTION: Let $E$ be an ellipse with centre at origin $O$ and the major and minor axis to be $2a$ and $2b$ respectively. Let $\theta$ be the acute angle at which $E$ is cut by a circle with centre ...
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How to construct a tangent to a hyperbola that is parallel to a given line? [closed]

You are given a hyperbola $h$, its asymptotes and its foci. You are also given some line $p$. Construct the line(s) tangent to $h$ and parallel to $p$. This problem came up while I was doing ...
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Prove that the normal to the parabola at $Q$ bisects the angle $FQP$

QUESTION: Consider the parabola $y^2=4x$. Let $P=(a,b)$ be any point inside the parabola, i.e. $b^2<4a$ and let $F$ be the focus of the parabola. Find the point $Q$ on the parabola such that $FQ+QP$...
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Tangent line Question

I am stumped with this question. Find the line tangent $f(t)=3\sin(2t)+5$ at the point where $t=\pi$. You must first find the derivative of $f$ at $\pi$. Next, find the equation of the tangent line ...
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Tangent to a point for a graph $t, f(t), g(t)$ is a diagonal to a box.

Consider a graph $t, x, y$ where $x=f(t)$ and $y=g(t) $ The tangent to a point $t, f(t), g(t)$ is a diagonal to the box with sides $dt, dx$, and $dy$. How is that so?
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Tangent of an Ellipse [closed]

For the curve described by this expression answer the following questions: $$x^2 + xy + y^2 = 7$$ Find the equation(s) of all lines tangent to the curve at $x = -1$. Give line in slope-intercept ...
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1answer
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Find the equations of two lines through the origin that are tangent to the ellipse equation: $2 {x^2} - 4 x + {y^2} + 1 = 0$

The answer is given. It is equal to $y = x \sqrt{2}$ and $y = -x \sqrt{2}$. Can you help me solve it?
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Prove that $PR$ tangents to the incircle of $\triangle{ABC}$

Incircle $(I)$ of $\triangle{ABC}$ tangents to $AB$ and $AC$ at $M$ and $N$ respectively. Let $P$ be any point lie on $BC$. $AP$ cuts $CM$ at $Q$. $NQ$ cuts $AB$ at $R$. Prove that $PR$ tangents to ...
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1answer
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Help: Point where a vector is normal to a surface?

In my problem set I have the following kind of exercises: I am given a normal vector and a surface $F(x,y,z)=0$ and $N=(a,b,c)$ the normal vector. I am asked to determine the points where the vector ...
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Calculate the coordinates of the smaller circle

In the image below, the larger circle is centered on the origin (0,0). The two circles are tangent and of known radius. The blue line is tangent to the larger circle and passes through the point of ...
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Find the coordinates of T (point where tangent touches the circle).

Given that P (a point that lies on the tangent) $= (6,-6)$ and the equation of the circle is $(x+5)^2 + (y-4)^2 = 25$. I'm unsure about my answer to this question and would like to know what answers ...
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3answers
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Two overlapping circles with tangents drawn at their intersection points intersecting at each others' centres.

So I'm stumped by what should be a rather simple problem. There are two circles whose tangents intersect at each others' centres. The tangents are at right angle. If I know the distance between the ...
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3answers
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How do you find an equation of the tangent line to the parabola

Determine the equation of the line that is tangent to the parabola with equation $y = x^2 − 2x + 2$ at the point $(3, 5)$
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Calculus A Level Equation Of Normal To Curve Defined in Parametric Form with Trig Functions

guys how are you doing today? Can you take some time to look over this equation of a curve defined in terms of $\theta $ and suggest how I can find the equation of the normal at the point $(a, \frac{a\...
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2answers
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Calculus A Level Line Tangent to Circle

How can you find values of $k$ such that $y = kx + 1$ is tangent to the circle $(y-1)^2 + (x-5)^2 = 9 $? I first rewrote the circle equation in terms of y: $$ (y-1)^2 = -(x-5)^2 + 9 \\y-1 = \pm\...
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1answer
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If the graph of an equation intersects the x-axis, is it possible for there to be a horizontal tangent

I would add a picture of the equation that this question pertained to, but the file size is too large The equation is $x^2 + 2x + y^4 + 4y = 5$. The question was "Is it possible for this curve to ...
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Finding a tangent plane equation

I have to find the tangent plane equation to the surface $zx^2+xy^2+yz^2=5$ at the point of $(-1,1,2)$. I couldn't get the right answer.
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Find equation of two tangent lines to ellipse $x^2+4y^2=36$ drawn from $(12,3)$ [duplicate]

Find the equation of the two tangent lines to the ellipse $x^2+4y^2=36$ that pass through the point $(12,3)$. I tried using implicit differentiation, and then I didn't know where to go from there.
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1answer
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Pre-Calculus Question: How to find the line of sight that someone cannot see?

If Dave is standing next to a silo of cross-sectional radius r=9 feet at the indicated position, his vision will be partially obstructed. Find the portion of the y-axis that Dave cannot see. (Hint: ...
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A symbol/notation for a line tangent to something?

I was doing problems of analytical geometry, specifically things like: "Find the equation of the circumference given the line tangent to it of equation...". I was wondering, since there is a way to ...
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1answer
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Is $x^2$ function differentiable at $x=0$?

My book says that $x^2$ function is differentiable at $x=0$. How is this possible, given that the right limit is greater than zero and the left limit is less than zero?
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1answer
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Defining intersection of two surfaces

I am having a bit of trouble beginning this question. The question is as follows: Let $r$ be the curve which is the intersection of the surface $z = \frac{1}{3}x^2 + \frac{2}{3}y^2$ and the surface: ...
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4answers
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A curve has the parametric equations $x=2t^2$ and $y=4t$. What is the value(s) of $k$ such that $y=x+k$ is a tangent to the curve?

A curve has the parametric equations $x=2t^2$ and $y=4t$. Find the value(s) of $k$ such that $y=x+k$ is a tangent to the curve. I get that you need to use differentiation to do this and I've tried ...
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How do I find the unittangentvector to given the parametric equation to the curve?

I need some help with understanding this task. A cycloid $C:[0,2\pi]\rightarrow R^2$ is given by $$ C = \begin{cases} x(t)=b(t-sint) \\ y(t)=b(1-cost) \end{cases} $$ Find the ...
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Consider the circles $x^2+y^2=1$ and $x^2+y^2-2x-6y+6=0$. Find the equations of common tangents to the two circles.

Applying condition of tangency $$y=mx\pm \sqrt {1+m^2}$$ $$y-3=m(x-1)\pm 2\sqrt{1+m^2}$$ So $$\pm \sqrt {1+m^2}=3-m\pm 2\sqrt{1+m^2}$$ $$3-m=\pm \sqrt{1+m^2}$$ $$m=\frac 43$$ So there should be ...
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How to define tangential (circumferential) angle on a sphere

I have a cross section which is shown below: I have made hemisphere by revolving the above cross section around Y axis as shown here: My question is: How can I define tangential angle in Cartesian ...
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if four circles touch each other externally then points of contact are concyclic

Question - if four circles S1,S2,S3,S4 touch each other externally then points of contact A,B,C,D are concyclic... Figure - My proof - First I invert about A and I get two parallel lines S1' ...
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How do I determine the x-coordinates of the points of intersection of a tangent line and the parabola y = x^2?

So in the following problem I have completed part a, but I am stuck on part b. I am trying to find the points of intersection of the tangent line I found in part a and the parabola y = x^2. However, I ...
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1answer
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Find Equation of a tangent of a Trig graph given domain of x and angle

Usually I am able to find the equation of the tangent when given at least the x point... But in case, I just got the domain. This is my problem of finding the ...
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2answers
43 views

incircle tanget to triangle at D and incirles of ADC ADB

$ABC$ is a triangle the circle is tangent to $BC$ at $D$ prove that the incircles of $\triangle ABD$ and $\triangle ACD$ are tangent to each other. What i tried is calling the smaller incircles ...
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Tangent Line at an Irrational Point on a Curve

I can construct an irrational point $P$ geometrically which is irrational due to square root, since, from a unit length straightedge and compass constructions we can only create lengths consisting of ...

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