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GeometryQuestion and Answers: Page 48

Question Number 147009    Answers: 1   Comments: 0

a parabola y=x^2 −15x+36 cuts the x axis at P and Q. a circle is drawn through P and Q so that the origin is outside it. then find the length of tangent to the circle from (0,0)?

$${a}\:{parabola}\:{y}={x}^{\mathrm{2}} −\mathrm{15}{x}+\mathrm{36}\:{cuts}\:{the}\: \\ $$$${x}\:{axis}\:{at}\:{P}\:\:{and}\:{Q}.\:{a}\:{circle}\:{is}\:{drawn} \\ $$$${through}\:{P}\:{and}\:{Q}\:{so}\:{that}\:{the}\:{origin} \\ $$$${is}\:{outside}\:{it}.\:{then}\:{find}\:{the}\:{length}\: \\ $$$${of}\:{tangent}\:{to}\:{the}\:{circle}\:{from}\:\left(\mathrm{0},\mathrm{0}\right)? \\ $$

Question Number 146519    Answers: 0   Comments: 0

Question Number 146494    Answers: 0   Comments: 0

Solid cube ABCD.EFGH is cut by a plane X so that it forms a plane of slices IJKLMN where I is mid AB, J is mid BF K is mid FG, L is mid HG, M mid DH and N mid AD. If the edge of the cube is X, find the area of the IJKLMN field

$$ \\ $$$$\mathrm{Solid}\:\mathrm{cube}\:\mathrm{ABCD}.\mathrm{EFGH}\:\mathrm{is}\:\mathrm{cut}\:\mathrm{by}\:\mathrm{a} \\ $$$$\mathrm{pla}{ne}\:{X}\:\mathrm{so}\:\mathrm{that}\:\mathrm{it}\:\mathrm{forms}\:\mathrm{a}\:\mathrm{plane}\:\mathrm{of}\:\mathrm{slices}\: \\ $$$$\mathrm{IJK}{LM}\mathrm{N}\:\mathrm{where}\:\mathrm{I}\:\mathrm{is}\:\mathrm{mid}\:\mathrm{AB},\:\mathrm{J}\:\mathrm{is}\:\mathrm{mid}\:\mathrm{BF} \\ $$$$\:{K}\:\mathrm{is}\:\mathrm{mid}\:\mathrm{FG},\:\:\mathrm{L}\:\mathrm{is}\:\mathrm{mid}\:\mathrm{HG},\:\:\mathrm{M}\:\mathrm{mid}\:\mathrm{DH}\: \\ $$$$\mathrm{and}\:\mathrm{N}\:\mathrm{mid}\:\mathrm{AD}.\:\mathrm{If}\:\mathrm{the}\:\mathrm{edge}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cube}\: \\ $$$$\mathrm{is}\:{X},\:\:\mathrm{find}\:\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\mathrm{the}\:\mathrm{IJKLMN}\:\mathrm{field} \\ $$

Question Number 146487    Answers: 0   Comments: 0

Cube ABCD.EFGH has side length a. Point P lies on AC such that AP : PC = 3 : 1 Through P a line l parallel to BD is drawn such that l each intersect BC at X and CD at Y. If AC and BD intersect at O find the distance between XY with OG!

$$ \\ $$$$\mathrm{Cube}\:\mathrm{ABCD}.\mathrm{EFGH}\:\mathrm{has}\:\mathrm{side}\:\mathrm{length}\:\mathrm{a}. \\ $$$$\mathrm{P}{o}\mathrm{int}\:\mathrm{P}\:\mathrm{lies}\:\mathrm{on}\:\mathrm{AC}\:\mathrm{such}\:\mathrm{that}\:\mathrm{AP}\::\:\mathrm{PC}\:=\:\mathrm{3}\::\:\mathrm{1}\:\mathrm{Through}\:\mathrm{P}\:\mathrm{a}\:\mathrm{line}\:\mathrm{l}\:\mathrm{parallel}\:\mathrm{to}\:\mathrm{BD}\:\mathrm{is} \\ $$$$\mathrm{dr}{a}\mathrm{wn}\:\mathrm{such}\:\mathrm{that}\:\mathrm{l}\:\mathrm{each}\:\mathrm{intersect}\:\mathrm{BC}\:\mathrm{at}\: \\ $$$$\mathrm{X}\:\mathrm{and}\:\mathrm{CD}\:\mathrm{at}\:\mathrm{Y}.\:\mathrm{If}\:\mathrm{AC}\:\mathrm{and}\:\mathrm{BD}\: \\ $$$$\mathrm{intersect}\:\mathrm{at}\:\mathrm{O}\:\mathrm{find}\:\mathrm{the}\:\mathrm{distance}\: \\ $$$$\mathrm{between}\:\mathrm{XY}\:\mathrm{with}\:\mathrm{OG}! \\ $$

Question Number 146048    Answers: 1   Comments: 0

in a triangle ABC we have { ((3sinA^ +4cosB^ =6)),((4sinB^ +3cosA^ =1)) :} find C^

$${in}\:{a}\:{triangle}\:{ABC}\:\:{we}\:{have}\: \\ $$$$\begin{cases}{\mathrm{3}{sin}\hat {{A}}+\mathrm{4}{cos}\hat {{B}}=\mathrm{6}}\\{\mathrm{4}{sin}\hat {{B}}+\mathrm{3}{cos}\hat {{A}}=\mathrm{1}}\end{cases} \\ $$$${find}\:\hat {{C}} \\ $$$$ \\ $$

Question Number 146000    Answers: 0   Comments: 3

Question Number 145575    Answers: 2   Comments: 0

Question Number 145573    Answers: 1   Comments: 0

Question Number 145379    Answers: 1   Comments: 1

Question Number 145370    Answers: 0   Comments: 0

There are two circles , C of radius 1 and C_r of radius r which intersect on a plain At each of the two intersecting points on the circumferences of C and C_r ,the tangent to C and that to C_r form an angle 120° outside of C and C_r . Fill in the blanks with the answers to the following questions (1) Express the distance d between the centers of C and C_r in terms of r (2) Calculate the value of r at which d in (1) attains the minimum (3) in case(2) express the area of the intersection of C and C_r in terms of the constant π

$$\mathrm{There}\:\mathrm{are}\:\mathrm{two}\:\mathrm{circles}\:,\:\mathrm{C}\:\mathrm{of}\:\mathrm{radius}\:\mathrm{1}\:\mathrm{and}\:\mathrm{C}_{\mathrm{r}} \: \\ $$$$\mathrm{of}\:\mathrm{radius}\:\mathrm{r}\:\mathrm{which}\:\mathrm{intersect}\:\mathrm{on}\:\mathrm{a}\:\mathrm{plain}\: \\ $$$$\mathrm{At}\:\mathrm{each}\:\mathrm{of}\:\mathrm{the}\:\mathrm{two}\:\mathrm{intersecting} \\ $$$$\mathrm{points}\:\mathrm{on}\:\mathrm{the}\:\mathrm{circumferences}\:\mathrm{of} \\ $$$$\mathrm{C}\:\mathrm{and}\:\mathrm{C}_{\mathrm{r}} \:,\mathrm{the}\:\mathrm{tangent}\:\mathrm{to}\:\mathrm{C}\:\mathrm{and} \\ $$$$\mathrm{that}\:\mathrm{to}\:\mathrm{C}_{\mathrm{r}} \:\mathrm{form}\:\mathrm{an}\:\mathrm{angle}\:\mathrm{120}°\:\mathrm{outside} \\ $$$$\mathrm{of}\:\mathrm{C}\:\mathrm{and}\:\mathrm{C}_{\mathrm{r}} .\:\mathrm{Fill}\:\mathrm{in}\:\mathrm{the}\:\mathrm{blanks}\: \\ $$$$\mathrm{with}\:\mathrm{the}\:\mathrm{answers}\:\mathrm{to}\:\mathrm{the}\:\mathrm{following} \\ $$$$\mathrm{questions}\: \\ $$$$\left(\mathrm{1}\right)\:\mathrm{Express}\:\mathrm{the}\:\mathrm{distance}\:\mathrm{d}\:\mathrm{between} \\ $$$$\mathrm{the}\:\mathrm{centers}\:\mathrm{of}\:\mathrm{C}\:\mathrm{and}\:\mathrm{C}_{\mathrm{r}} \:\mathrm{in}\:\mathrm{terms} \\ $$$$\mathrm{of}\:\mathrm{r}\: \\ $$$$\left(\mathrm{2}\right)\:\mathrm{Calculate}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{r}\:\mathrm{at}\: \\ $$$$\mathrm{which}\:\mathrm{d}\:\mathrm{in}\:\left(\mathrm{1}\right)\:\mathrm{attains}\:\mathrm{the}\:\mathrm{minimum} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{in}\:\mathrm{case}\left(\mathrm{2}\right)\:\mathrm{express}\:\mathrm{the}\:\mathrm{area} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{intersection}\:\mathrm{of}\:\mathrm{C}\:\mathrm{and}\:\mathrm{C}_{\mathrm{r}} \\ $$$$\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{the}\:\mathrm{constant}\:\pi \\ $$

Question Number 145246    Answers: 1   Comments: 0

prove that a triangle inscribed in a circle of radius r having maximum area is an equilateral triangle with side (√3)r.

$$\mathrm{prove}\:\mathrm{that}\:\mathrm{a}\:\mathrm{triangle}\:\mathrm{inscribed}\: \\ $$$$\mathrm{in}\:\mathrm{a}\:\mathrm{circle}\:\mathrm{of}\:\mathrm{radius}\:\mathrm{r}\:\mathrm{having}\:\mathrm{maximum} \\ $$$$\mathrm{area}\:\mathrm{is}\:\mathrm{an}\:\mathrm{equilateral}\:\mathrm{triangle}\:\mathrm{with} \\ $$$$\mathrm{side}\:\sqrt{\mathrm{3}}\mathrm{r}. \\ $$

Question Number 145244    Answers: 0   Comments: 0

consider the circle (x−1)^2 +(y−1)^2 =2, A(1,4), B(1,−5). if P is a point on the circle such that PA+PB is maximum then prove that P,A,B are collinear points.

$$\mathrm{consider}\:\mathrm{the}\:\mathrm{circle}\: \\ $$$$\left(\mathrm{x}−\mathrm{1}\right)^{\mathrm{2}} +\left(\mathrm{y}−\mathrm{1}\right)^{\mathrm{2}} =\mathrm{2}, \\ $$$$\mathrm{A}\left(\mathrm{1},\mathrm{4}\right),\:\mathrm{B}\left(\mathrm{1},−\mathrm{5}\right).\:\mathrm{if}\:\mathrm{P}\:\mathrm{is}\: \\ $$$$\mathrm{a}\:\mathrm{point}\:\mathrm{on}\:\mathrm{the}\:\mathrm{circle}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{PA}+\mathrm{PB}\:\mathrm{is}\:\mathrm{maximum}\:\mathrm{then} \\ $$$$\mathrm{prove}\:\mathrm{that}\:\mathrm{P},\mathrm{A},\mathrm{B}\:\mathrm{are}\:\mathrm{collinear}\: \\ $$$$\mathrm{points}. \\ $$

Question Number 144335    Answers: 0   Comments: 0

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Question Number 144332    Answers: 0   Comments: 0

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Question Number 144268    Answers: 2   Comments: 0

Question Number 143463    Answers: 0   Comments: 0

Π_(k=1) ^n tan(((kπ)/(2n+1)))=(√(2n+1))

$$\underset{{k}=\mathrm{1}} {\overset{{n}} {\prod}}{tan}\left(\frac{{k}\pi}{\mathrm{2}{n}+\mathrm{1}}\right)=\sqrt{\mathrm{2}{n}+\mathrm{1}} \\ $$

Question Number 143168    Answers: 2   Comments: 2

Question Number 142849    Answers: 1   Comments: 0

Question Number 142560    Answers: 1   Comments: 0

Question Number 142546    Answers: 0   Comments: 1

i need help

$${i}\:{need}\:{help} \\ $$

Question Number 142282    Answers: 1   Comments: 0

Three interior angles of a polygon are 160° each. If the other interior angles are 120° each, find the number of sides of the polygon.

$$\mathrm{Three}\:\mathrm{interior}\:\mathrm{angles}\:\mathrm{of}\:\mathrm{a}\:\mathrm{polygon}\:\mathrm{are}\:\mathrm{160}° \\ $$$$\mathrm{each}.\:\mathrm{If}\:\mathrm{the}\:\mathrm{other}\:\mathrm{interior}\:\mathrm{angles}\:\mathrm{are}\:\mathrm{120}°\:\mathrm{each}, \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{sides}\:\mathrm{of}\:\mathrm{the}\:\mathrm{polygon}. \\ $$

Question Number 142306    Answers: 0   Comments: 0

Question Number 142164    Answers: 0   Comments: 1

Question Number 141983    Answers: 0   Comments: 1

Question Number 141750    Answers: 1   Comments: 0

Question Number 142218    Answers: 1   Comments: 0

Straight line lx+my=1 is tangent to the curve (ax)^n +(by)^n =1 Prove that ((l/a))^(n/(n−1)) +((m/b))^(n/(n−1)) =1

$${Straight}\:{line}\:{lx}+{my}=\mathrm{1}\:\:{is}\:{tangent}\:{to}\:{the}\:{curve}\:\left({ax}\right)^{{n}} +\left({by}\right)^{{n}} =\mathrm{1} \\ $$$${Prove}\:{that}\:\left(\frac{{l}}{{a}}\right)^{\frac{{n}}{{n}−\mathrm{1}}} +\left(\frac{{m}}{{b}}\right)^{\frac{{n}}{{n}−\mathrm{1}}} =\mathrm{1} \\ $$

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