Questions tagged [tangent-line]
For questions on the tangent line, the unique straight line that touches a function locally only once.
871
questions
3
votes
0answers
24 views
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 ...
1
vote
1answer
24 views
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 ...
1
vote
1answer
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 = - ...
-1
votes
2answers
20 views
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?
0
votes
0answers
28 views
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}
...
0
votes
2answers
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 \...
2
votes
2answers
38 views
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 ...
0
votes
4answers
34 views
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 ...
1
vote
1answer
16 views
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 ...
1
vote
3answers
41 views
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 ...
0
votes
2answers
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 ...
0
votes
2answers
29 views
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)...
0
votes
1answer
20 views
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 $M(c,f(c))$ ...
1
vote
1answer
72 views
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 ...
2
votes
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 ...
1
vote
1answer
11 views
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 ...
0
votes
0answers
15 views
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 ...
0
votes
2answers
47 views
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 ...
-1
votes
0answers
13 views
showing tangent line on differentiable funcions
]1
Can someone solve this?
Explain your answer in detail please.
0
votes
1answer
34 views
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 ...
0
votes
2answers
26 views
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 ...
0
votes
1answer
21 views
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 $FQ+QP$...
0
votes
3answers
37 views
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 ...
0
votes
0answers
13 views
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?
-1
votes
1answer
54 views
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 ...
0
votes
1answer
26 views
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?
0
votes
1answer
26 views
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 ...
1
vote
1answer
17 views
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 ...
0
votes
2answers
52 views
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 ...
1
vote
2answers
24 views
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 ...
2
votes
3answers
42 views
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 ...
0
votes
3answers
24 views
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 $(3, 5)$
0
votes
0answers
26 views
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\...
1
vote
2answers
34 views
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\...
2
votes
1answer
34 views
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 ...
-1
votes
1answer
22 views
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.
0
votes
3answers
45 views
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.
0
votes
1answer
36 views
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: ...
1
vote
0answers
16 views
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 ...
0
votes
1answer
52 views
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?
0
votes
1answer
25 views
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: ...
0
votes
4answers
38 views
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 ...
0
votes
0answers
8 views
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}
$$
Find the ...
0
votes
3answers
50 views
Consider the circles $x^2+y^2=1$ and $x^2+y^2-2x-6y+6=0$. Find the equations of common tangents to the two circles.
Applying condition of tangency
$$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 ...
0
votes
0answers
14 views
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 ...
0
votes
2answers
71 views
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' ...
0
votes
1answer
31 views
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 ...
0
votes
1answer
53 views
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 ...
1
vote
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 ...
0
votes
0answers
23 views
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 ...
