# 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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### Competition problem based on triangles [closed]

Let $ABC$ be a triangle with $AC > AB$, and denote its circumcircle by $\Omega$ and incentre by $I$. Let its incircle meet sides $BC, CA, AB$ at $D, E, F$ respectively. Let $X$ and $Y$ be two ...
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### ODE has many tangent solutions

This is perhaps one of the most over-asked questions on this site: say here, or here. However, the answers are not satisfactory, especially of the second assertion. The first assertion is ...
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### Find the equation of tangent through the point (3,4) on the circle $x^2+y^2 = 9$

radius = 3 Let $y-y_1 = m(x-x_1)$ be the equation of tangent. Since, tangent passes through (3,4) $or,\text{ }y - 4 = m(x - 3)$ $or,\text{ }y - 4 = mx - 3m$ $or,\text{ }mx +y - 3m + 4 = 0$ Thus, ...
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### Is there a more geometrical definition of the tangent line of a curve? based on the intuitive idea that a tangent line only touches at one point

I asked this question to my calculus teacher and it was a frustrating experience, basically he would say, over and over again, that a tangent line at a point is a line that goes through that point and ...
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### Calculating angle for line tangent to circle through a point

I have a circle of fixed radius $r$. I have a target that is $x$ units laterally separated from the center of the circle, and $y$ units vertically. I need to calculate the angle $θ$ which is the ...
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### Given a circle and an external point, find the x intercept of the line tangent to the circle and goes through the point.

The equation of the circle is given by $x^2+(y-r)^2=r^2$ where $r$ is the radius. The point is located at the point $(d,h)$. Here is my approach to this: the general equation of all lines that passes ...
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### how do i find the equations of L1 and L2 [closed]

L1 and L2 are perpendicular. the equation of the circle is given as $x^2+6x+y^2-2y=7$. line L1 cuts the circle at $P$. L2 cuts the circle at $Q$. I need to find the equations of the lines L1 and L2. I ...
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### Find center of externally tangent circle

I've been struggling to find a way to resolve the following problem: Let $C_1$ a circle of center $V$ and of radius $r_1$. Let $A$ and $B$ two points outside of $C_1$, and $L$ a line passing by them. ...
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1 vote
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### How do I find a constant $C$, when $x = a$, through the tangent line? [closed]

The full question is as follows: For some constant $C$, the equation of the tangent line to the graph of $y = f(x)$ = $4x^4$+C at the point where $x=a$ is $y = −78.608x − 106.1252$ Find $C$. My ...
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### Finding Equations for Common Tangents Between Two Ellipses, A General Solution

I'm interested in finding the four common tangents between two ellipses. While I've found some fascinating approaches using dual conics to identify their intersection points (link 1, link 2), my ...
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### Is it correct to say that if two parabolas touch, then tangents at the point of intersection have the same slope?

Is it correct to say that if two parabolas touch, then tangents at the point of intersection have the same slope? If it is true, is there a neat geometric interpretation of the fact?
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### Circle tangent to rotated ellipse and horizontal line

I would like to find the position for the center of a circle $(x_0, y_0)$ that is tangent to both an ellipse and a horizontal line. The ellipse is positioned at $(0,0)$ and is defined by major axis $a$...
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### Point of contact between an ellipse $(5 \cos t , 3 \sin t)$ and an apparent tangent $x \cos(R) + y \sin(R) =D$ sliding on it.

Let $E$ be an ellipse defined by $(x^2 / 5^2) + ( y^2 / 3^2) = 1$ or, equivalently $( 5\cos t , 3\sin t )$ with $0\leq t \leq 2 \pi$. Let $P= ( 5 \cos R , 3 \sin R) \space 0\leq R \leq 2 \pi$ be a ...
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### Why is a derivative undefined at its discontinuities?

This question deals with why the derivative of $f$ is not defined at discontinuities in $f$. I found the answers satisfactory. My question deals with why the derivative is not defined at ...
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### Given points $A$, $B$, $C$ and $D$ lying on a circle and lines $BD$ and $AC$ intersecting at $F$, prove lines are parallel

The diagram shows the points $A$, $B$, $C$ and $D$ lying on a circle. $AC$ and $BD$ intersect at point $F$. $EG$ is tangent to the circle at point $C$. $AD$ is produced to meet the tangent at point $E$...
1 vote
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### Find the equation of the line passing through the 2 points of the tangents on a circle from point $(x_1,y_1)$

Given a circle $x^2+y^2=r^2$ and a point P(x1,y1) outside of the circle. I can draw two tangents from P to the circle. I will call A and B the points where the tangents cross the circle. How can I ...
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### AB is an external tangent to $S_1$ at A and to $S_2$ at B and common tangent at P cuts AB at Q.

Question: Two circles $S_1$ and $S_2$ of radius $3$ and $4$ touch each other externally at point P. If AB is an external tangent to $S_1$ at A and to $S_2$ at B and common tangent at P cuts AB at Q. ...
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### Are the tangent lines at the farthest-separated points on a closed plane curve always parallel?

Suppose you have a closed differentiable plane curve. Are the tangent lines to the curve at the most distant points on the curve always parallel? What if we assume that the curve is convex? I don't ...
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### How to find the tangent lines of a parabola which must pass through a point without calculus. [closed]

I‘m tutoring a student who doesn’t know calculus (yet). I was given the following question: Find all lines of tangency of the graph $y=x^2$ which pass through the point $P(-6,-5)$. I know I have to ...
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### If the length $BC=l$, length of arc $AB=l_1$ and length of arc $AC=l_2$, then $l+l_1+l_2=$

Question: A circle with centre $C_1$ and radius $\frac32$ touches another circle with centre $C_2$ and radius $\frac12$ externally at point $A$. A common tangent touches circle with centre $C_1$ at B ...
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