Circle center : Cx,Cy  Circle radius : a  Point from which we need to draw a tangent line : Px,Py  I need the formula to find the two tangents (t1x, t1y) and (t2x,t2y) given all the above.
122402
I am given the equation of a circle: $(x + 2)^2 + (y + 7)^2 = 25$. The radius is $5$. Center of the circle: $(-2, -7)$.  Two lines tangent to this circle pass through point $(4, -3)$, which is outside of said circle. How would I go about finding one of the equations of the lines tangent to the circle?  I haven't started calculus, so I ask for advice fitting for someone starting a topic like this.
108111
Why is $\pi r^2$ the surface of a circle?   I have learned this formula ages ago and I'm just using it like most people do, but I don't think I truly understand how circles work until I understand why this formula works.   So I want to understand why it works and not just how.  Please don't use complicated symbols.
3557
Let $\Gamma$ be a circle that passes through the origin. Show that we can find real numbers $s$ and $t$ such that $\Gamma$ is the graph of $r = 2s \cos (\theta + t).$  I know this has to be converted to a cartesian equation, but how do I do this, and what do I do after?  Thanks
116593
A circle has the same center as an ellipse and passes through the foci $F_1$ and $F_2$ of the ellipse,such that the two curves intersect in $4$ points.Let $P$ be any one of their point of intersection.If the major axis of the ellipse is $17$ and the area of the triangle $PF_1F_2$ is $30$,then find the distance between the foci.    Let the center of the ellipse and the circle be $(0,0)$ We are given $2a=$length of major axis$=17$. Let the coordinates of foci be $F_1(c,0)$ and $F_2(-c,0)$ We need to find $2c$. Area of $PF_1F_2=\frac{1}{2}\times 2c\times$perpendicular distance between $P$ and the axis of the major axis of the ellipse. I do not know how to solve it further.
5529
Given the radius and $x,y$ coordinates of the center point of two circles how can I calculate their points of intersection if they have any?
87782
Let $u=(6,7), v=(-2, 3), w=(7,0)$. Find the center and the radius of the circumscribed circle of triangle $[u,v,w]$.  My approach would look like this:  (1) Determine the angles of the triangle $[u,v,w]$ via dot products.  (2) If we draw three lines from the the center $o$ of the circle to the three vertices, the triangle $[u,v,w]$ will be made of three equilateral triangles. We can then easily find from (1) all the nine angles of the three triangles.  (3) Determine the heights of the three inscribed triangles, which will be the same for all of the three, and which will be the radius $r$ of the inscribed circle.  (4) Then we can drop a perpendicular line from $o$ to, say, $[v,w]$ in order to find the coordinates of $o$.  But is there a much less tedious way to solve this problem? And do you think my approach is correct?
65289
No idea how to do this, I used to have these conic shapes committed to memory but I forget them already.   I am supposed to find an equation for the circle that has center $(-1, 4)$ and passes through $(3, -2)$.  I tried graphing it, but it didn't help since the point was not in a straight line with the center and I have no idea what the diameter is.
28090
Suppose you are given 3 circles A,B,C such that A is tangent to B and B is tangent to C. How could you construct a circle (not necessarily all of them) that is tangent to all 3? (If such a circle exists)  I have developed a method of constructing a circle tangent to two tangent circles (pick a point P on their radical axis, draw the tangents from that point to both circles, for each circle construct the line joining the circle center and the tangency point from P, intersect those lines, and draw the circle centered in this intersection and which contains one of the tangency points from P) but I can not figure out how to adapt this idea for 3 circles.
76114
The parametric equation  $$\begin{align*} x(t) &amp;= \cos t\\ y(t) &amp;= \sin t \end{align*}$$  traces the unit circle centered at the origin ($x^2+y^2=1$).   Similarly,   $$\begin{align*} x(t) &amp;= \cosh t\\ y(t) &amp;= \sinh t \end{align*}$$  draws the right part of a regular hyperbola ($x^2-y^2=1$).  The hyperbolic trigonometric functions are very similar to the standard trigonometric function.  Do similar functions exist that trace parabolas (because it is another conic section) when set up as parametric equations like the above functions?  If so, are they also similar to the standard and hyperbolic trigonometric functions?
11395
I have two function $f$ and $r$, such that: $y = f(x)$, $r(y) = x$. I read that it means that: $f \circ r = I_{U}$, where $x \in U, y \in U$, where $U$ is a finite set. What does $f \circ r = I_{U}$ mean? What does this circle mean and how could i read all this writing $f \circ r = I_{U}$ ?
101216
Let $A_1 A_2 \dotsb A_{11}$ be a regular $11$-gon inscribed in a circle of radius $2$. Let $P$ be a point, such that the distance from $P$ to the center of the circle is $3$. Find $$PA_1^2 + PA_2^2 + \dots + PA_{11}^2$$  This kind of reminded me of roots of unity, because of a regular polygon inscribed in a circle, but I'm not sure on how to work this problem.
70038
A Triangle has 2 sides of length 4 and 9. The largest circle than can be drawn in the triangle so as to touch all 3 sides has radius 1 cm. Find the length of the 3rd side.
66625
Let $A$ and $B$ be two fixed points on a fixed straight line. Two circles touch this line at $A$ and $B$ respectively and tangent to each other at $M$. When the circles vary, what does the locus of $M$ form?
44621
I'm trying to figure out how to calculate the following angle:  I know center (p1) and radius (r) of a circle. Given a point p3 I want to calculate the angle a such as the tangent (tan) of the circle at point p4, points in p3 direction.  Here's a figure:    Doesn anyone know how to do that?
95238
While browsing around about problems similar to , I have found references to constructions of all types of circles. For example, not only is it possible to construct a circle tangent to three given circles, but one can construct a circle through any three points, tangent to any three lines, passing through two given points and tangent to a line or circle, passing through a given point and tangent to two given lines or circles, etc. Pretty much any combination of criteria regarding points, lines, and circles, with repetition.  Can we have one of each? I haven't found any resource saying whether it's possible or not to construct a circle through a given point, tangent to a given line, and tangent to a given circle. Is such a construction possible? Thanks.
117503
Suppose $ T \subset \mathbb{C} $. Show that the corresponding set $ S \subset \Sigma $ is  a. a circle if $ T $ is a circle. b. a circle minus (0, 0, 1) if $ T $ is a line.  Here we are defining $ \Sigma $ to be the Riemann sphere, given by the set: $$ \Sigma = \left \{(\xi, \eta, \zeta) : \xi^{2} + \eta^{2} + (\zeta - \frac{1}{2})^{2} = \frac{1}{4} \right \} $$  To take a point from $ \mathbb{C} $ to $ \Sigma $ we can use the following:  $$ \xi = \frac{x}{x^{2} + y^{2} + 1}; \eta = \frac{y}{x^{2} + y^{2} + 1}; \zeta = \frac{x^{2} + y^{2}}{x^{2} + y^{2} + 1} $$  We define a circle on $ \Sigma $ to be the intersection of a plane of the form $ A\xi + B\eta + C\zeta = D $ with $ \Sigma $. We also know the converse of this problem is true, that the intersection above yeilds a set in $ \mathbb{C} $ with the following property:  $ (C - D)(x^{2} + y^{2}) + Ax + By = D $. As you can see, when C = D, then an equation for a line is yeilded, otherwise it is a circle.   I really am at a  loss about how to solve this problem. The only thing I can think to do is to pick 3 points on a circle or radius $ r $ with center $ z_{0} $, use these points to find two vectors in $ \Sigma $, take their cross product to get a normal vector, use this normal vector to get a plane. Once I have the plane in form $ A\xi + B\eta + C\zeta = D $ then I could prove that the circle I had chosen corresponds exactly with $ (C - D)(x^{2} + y^{2}) + Ax + By = D $. Is there not an easier, less computation way to do this?
25357
Get the equation of a circle through the points $(1,1), (2,4), (5,3) $.  I can solve this by simply drawing it, but is there a way of solving it (easily) without having to draw?