# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
Determine the equation of the line that is tangent to the parabola with equation $y = x^2 − 2x + 2$ at the point $(3, 5)$