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  • ...>. (One needs to take some care when <math>P=Q</math> or when this line is tangent to the curve, and hence cuts it in only two points.) We define <math>P+Q</m A modular form <math>f</math> is a function such that for all <math>z\in\mathcal{H}</math> and all <math>\gamma\in SL_2
    5 KB (849 words) - 15:14, 18 May 2021
  • ...irst quadrant can be computed using a definite integral from 0 to r of the function <math>f(x) = \sqrt{r^2 - x^2}</math>. Using the substitution <math>x = r \s ...angent]] of that circle. Note that any point on a circle can have only one tangent.
    9 KB (1,585 words) - 12:46, 2 September 2024
  • ...h>3-4-5</math> right triangle are the centers of three mutually externally tangent circles, as shown. What is the sum of the areas of the three circles? ...e radii <math>3</math> and <math>8</math>, respectively. A common internal tangent intersects the circles at <math>C</math> and <math>D</math>, respectively.
    15 KB (2,223 words) - 12:43, 28 December 2020
  • Let <math>f</math> be a function with the following properties: ...meter <math>\overline{AB}</math> is constructed inside the square, and the tangent to the semicircle from <math>C</math> intersects side <math>\overline{AD}</
    13 KB (1,953 words) - 23:31, 25 January 2023
  • <math> \textbf{(A)}\ \text{a parabola tangent to the }x\text{-axis} </math> <math> \textbf{(B)}\ \text{a parabola not tangent to the }x\text{-axis}\qquad\textbf{(C)}\ \text{a horizontal line} </math>
    13 KB (1,955 words) - 20:06, 19 August 2023
  • ...tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the ...y. What is the length of the shortest line segment <math>PQ</math> that is tangent to <math>C_1</math> at <math>P</math> and to <math>C_2</math> at <math>Q</m
    12 KB (1,792 words) - 12:06, 19 February 2020
  • Let <math>f</math> be a [[function]] for which <math>f(x/3) = x^2 + x + 1</math>. Find the sum of all values o ...point <math>A</math>. The segment <math>AB</math> is [[tangent (geometry)|tangent]] to the circle at <math>A</math> and <math>\angle AOB = \theta</math>. If
    13 KB (1,948 words) - 09:35, 16 June 2024
  • Let <math>f</math> be a function satisfying <math>f(xy) = \frac{f(x)}y</math> for all positive real numbers ...math>B</math> with a radius of 4 are externally tangent. A third circle is tangent to the first two and to one of their common external tangents as shown. The
    13 KB (1,957 words) - 11:53, 24 January 2024
  • ...t <math>OX</math> be a radius of the larger circle, let <math>XZ</math> be tangent to the smaller circle at <math>Z</math>, and let <math>OY</math> be the rad A [[function]] <math>f</math> is defined by <math>f(z) = i\overline{z}</math>, where <ma
    13 KB (2,049 words) - 12:03, 19 February 2020
  • ...[[collinear]]. A [[chord]] of <math> C_3 </math> is also a common external tangent of <math> C_1 </math> and <math> C_2. </math> Given that the length of the ...e{O_1O_2}</math> at a point <math>H</math>. Let the endpoints of the chord/tangent be <math>A,B</math>, and the foot of the perpendicular from <math>O_3</math
    4 KB (693 words) - 12:03, 28 December 2021
  • ...six circles in the ring. Find <math> \lfloor K \rfloor</math> (the [[floor function]]).
    1 KB (213 words) - 12:17, 22 July 2017
  • ...h> and <math>B\,</math> on the larger circle, <math>\overline{CD}\,</math> tangent at <math>Q\,</math> to the smaller circle, and the smaller circle outside < The function <math>f_{}^{}</math> has the property that, for each real number <math>x,\,
    7 KB (1,141 words) - 06:37, 7 September 2018
  • Circles of radii 5, 5, 8, and <math>m/n</math> are mutually externally tangent, where <math>m</math> and <math>n</math> are relatively prime positive inte The function <math>f</math> defined by <math>f(x)= \frac{ax+b}{cx+d}</math>, where <math
    7 KB (1,098 words) - 16:08, 25 June 2020
  • ...where <math>a_{}</math> and <math>b_{}</math> are positive numbers. This function has the property that the image of each point in the complex plane is equid The inscribed circle of triangle <math>ABC</math> is tangent to <math>\overline{AB}</math> at <math>P_{},</math> and its radius is 21.
    7 KB (1,094 words) - 12:39, 16 August 2020
  • ...th>\overline{UV}</math> is perpendicular to <math>\overline{PQ}</math> and tangent to <math>C_{1}</math>. Let <math>C_{2}</math> be the inscribed circle of <m A certain function <math>f</math> has the properties that <math>f(3x) = 3f(x)</math> for all p
    8 KB (1,282 words) - 20:12, 19 February 2019
  • ...el to <math>BC</math>. This means that <math>BC</math> is parallel to the tangent to the given circle at <math>D</math>. ...ersection. By the problem condition, however, the circle <math>P</math> is tangent to <math>BC</math> at point <math>N</math>.
    20 KB (3,497 words) - 14:37, 27 May 2024
  • Since <math>\cot</math> is the reciprocal function of <math>\tan</math>: Using the tangent addition formula:
    3 KB (545 words) - 22:44, 12 October 2023
  • ...etry#Tangent|tangent]] function is <math>180^\circ</math>, and the tangent function is [[one-to-one]] over each period of its domain. ...180k</math> for some positive integer <math>k</math> (since the period of tangent is <math>180^{\circ}</math>), or <math>19 x \equiv 141 \pmod{180}</math>. N
    5 KB (757 words) - 20:59, 23 December 2024
  • ...The x-axis and the line <math>y = mx</math>, where <math>m > 0</math>, are tangent to both circles. It is given that <math>m</math> can be written in the form ...me positive reals <math>a</math> and <math>b</math>. These two circles are tangent to the <math>x</math>-axis, so the radii of the circles are <math>a</math>
    7 KB (1,188 words) - 16:46, 23 June 2024
  • The minimum value of the function ...</math> for [[real number|real]] <math>x</math>, we can now simplify the [[function]] to
    2 KB (331 words) - 23:37, 25 January 2023

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