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Question Number 109075 Answers: 1 Comments: 0

Given the equations of twe circles C_1 : x^2 + y^2 −6x−4y + 9 = 0 and C_2 : x^2 + y^2 −2x−6y + 9. (a) Find the equation of the circle C_3 which passes through the centre of C_1 and through the point of intersection of C_1 and C_2 . (b) The equations of two tangents from the origin to C_1 and the lenght of each tangent.

$$\mathrm{Given}\:\mathrm{the}\:\mathrm{equations}\:\mathrm{of}\:\mathrm{twe}\:\mathrm{circles}\: \\ $$$${C}_{\mathrm{1}} \::\:{x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} \:−\mathrm{6}{x}−\mathrm{4}{y}\:+\:\mathrm{9}\:=\:\mathrm{0}\:\mathrm{and}\:{C}_{\mathrm{2}} \::\:{x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} −\mathrm{2}{x}−\mathrm{6}{y}\:+\:\mathrm{9}. \\ $$$$\left(\mathrm{a}\right)\:\mathrm{Find}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{the}\:\mathrm{circle}\:{C}_{\mathrm{3}} \:\mathrm{which}\:\mathrm{passes}\:\mathrm{through}\:\mathrm{the}\:\mathrm{centre} \\ $$$$\mathrm{of}\:{C}_{\mathrm{1}} \:\mathrm{and}\:\mathrm{through}\:\mathrm{the}\:\mathrm{point}\:\mathrm{of}\:\mathrm{intersection}\:\mathrm{of}\:{C}_{\mathrm{1}} \:\mathrm{and}\:{C}_{\mathrm{2}} . \\ $$$$\left(\mathrm{b}\right)\:\mathrm{The}\:\mathrm{equations}\:\mathrm{of}\:\mathrm{two}\:\mathrm{tangents}\:\mathrm{from}\:\mathrm{the}\:\mathrm{origin}\:\mathrm{to}\:{C}_{\mathrm{1}} \:\mathrm{and}\:\mathrm{the}\:\mathrm{lenght} \\ $$$$\mathrm{of}\:\mathrm{each}\:\mathrm{tangent}. \\ $$

Question Number 109070 Answers: 0 Comments: 6

Question Number 109068 Answers: 2 Comments: 0

Question Number 109067 Answers: 1 Comments: 0

Question Number 109047 Answers: 1 Comments: 0

∫_(−2) ^∞ (x+2)^5 e^(−(x+2)) dx ∫_0 ^1 ((tan(x))/x)dx

$$\int_{−\mathrm{2}} ^{\infty} \left({x}+\mathrm{2}\right)^{\mathrm{5}} {e}^{−\left({x}+\mathrm{2}\right)} {dx} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{tan}\left({x}\right)}{{x}}{dx} \\ $$

Question Number 109029 Answers: 1 Comments: 0

Find the value(s) of a,b and c if: (x+a)(x+2020)+1=(x+b)(x+c) where a,b and c are natural numbers.

$${Find}\:{the}\:{value}\left({s}\right)\:{of}\:{a},{b}\:{and}\:{c}\:\:{if}: \\ $$$$\left({x}+{a}\right)\left({x}+\mathrm{2020}\right)+\mathrm{1}=\left({x}+{b}\right)\left({x}+{c}\right) \\ $$$${where}\:{a},{b}\:{and}\:{c}\:{are}\:{natural}\:{numbers}. \\ $$

Question Number 109022 Answers: 8 Comments: 0

Question Number 109018 Answers: 3 Comments: 0

Question Number 109017 Answers: 1 Comments: 0

Question Number 109016 Answers: 2 Comments: 0

Question Number 109005 Answers: 2 Comments: 0

i^i =?

$${i}^{{i}} =? \\ $$

Question Number 109004 Answers: 4 Comments: 0

^ Solve ∣3x+5∣ = ∣4x−3∣ where x ∈ R

$$\overset{} {\:}\:\:{Solve}\:\:\:\:\mid\mathrm{3}{x}+\mathrm{5}\mid\:=\:\mid\mathrm{4}{x}−\mathrm{3}\mid \\ $$$$\:\:\:{where}\:{x}\:\in\:\mathbb{R} \\ $$$$ \\ $$

Question Number 109001 Answers: 0 Comments: 0

Question Number 109000 Answers: 0 Comments: 0

Question Number 108999 Answers: 0 Comments: 0

Question Number 108998 Answers: 0 Comments: 0

Question Number 108997 Answers: 1 Comments: 0

Question Number 108987 Answers: 1 Comments: 0

Question Number 108984 Answers: 7 Comments: 1

B_≈ eM_≈ ath_≈ (1)Σ_(n = 1) ^(100) [ (2/(4n^2 −1)) ] ? (2) (2/(4.9)) + (2/(9.14)) + (2/(14.19)) + ... + (2/(49.54)) ? (3) Given a quadratic equation x^2 (Σ_(i = 1) ^2 2)+ x(Σ_(i = 1) ^5 i)−(2/3)(Σ_(i = 1) ^3 i) = 0 has the roots are x_1 and x_2 with x_1 < x_2 . Find the value of x_1 +8x_2 .

$$\:\:\:\underset{\approx} {\mathcal{B}}{e}\underset{\approx} {\mathcal{M}}{at}\underset{\approx} {{h}} \\ $$$$\left(\mathrm{1}\right)\underset{{n}\:=\:\mathrm{1}} {\overset{\mathrm{100}} {\sum}}\left[\:\frac{\mathrm{2}}{\mathrm{4}{n}^{\mathrm{2}} −\mathrm{1}}\:\right]\:? \\ $$$$\left(\mathrm{2}\right)\:\frac{\mathrm{2}}{\mathrm{4}.\mathrm{9}}\:+\:\frac{\mathrm{2}}{\mathrm{9}.\mathrm{14}}\:+\:\frac{\mathrm{2}}{\mathrm{14}.\mathrm{19}}\:+\:...\:+\:\frac{\mathrm{2}}{\mathrm{49}.\mathrm{54}}\:? \\ $$$$\left(\mathrm{3}\right)\:{Given}\:{a}\:{quadratic}\:{equation} \\ $$$${x}^{\mathrm{2}} \left(\underset{{i}\:=\:\mathrm{1}} {\overset{\mathrm{2}} {\sum}}\mathrm{2}\right)+\:{x}\left(\underset{{i}\:=\:\mathrm{1}} {\overset{\mathrm{5}} {\sum}}{i}\right)−\frac{\mathrm{2}}{\mathrm{3}}\left(\underset{{i}\:=\:\mathrm{1}} {\overset{\mathrm{3}} {\sum}}{i}\right)\:=\:\mathrm{0} \\ $$$${has}\:{the}\:{roots}\:{are}\:{x}_{\mathrm{1}} \:{and}\:{x}_{\mathrm{2}} \:{with}\: \\ $$$${x}_{\mathrm{1}} \:<\:{x}_{\mathrm{2}} .\:{Find}\:{the}\:{value}\:{of}\:{x}_{\mathrm{1}} +\mathrm{8}{x}_{\mathrm{2}} . \\ $$

Question Number 108967 Answers: 3 Comments: 0

b^★ o^★ b^★ h^∼ a^∼ n^∼ s^∼ 1+(4/(11))+(9/(121))+((16)/(1331))+((25)/(14641))+...

$$\:\:\overset{\bigstar} {{b}}\overset{\bigstar} {{o}}\overset{\bigstar} {{b}}\overset{\sim} {{h}}\overset{\sim} {{a}}\overset{\sim} {{n}}\overset{\sim} {{s}} \\ $$$$\mathrm{1}+\frac{\mathrm{4}}{\mathrm{11}}+\frac{\mathrm{9}}{\mathrm{121}}+\frac{\mathrm{16}}{\mathrm{1331}}+\frac{\mathrm{25}}{\mathrm{14641}}+... \\ $$

Question Number 108961 Answers: 3 Comments: 2

Solve ((∣x−2∣+1)/(∣x−2∣−1))<3

$$\mathrm{Solve}\:\frac{\mid{x}−\mathrm{2}\mid+\mathrm{1}}{\mid{x}−\mathrm{2}\mid−\mathrm{1}}<\mathrm{3} \\ $$

Question Number 108956 Answers: 2 Comments: 0

b^★ o^★ b^★ h^□ a^□ n^□ s^♠ lim_(x→0) ((tan 8x−sin 4x.cos 4x)/(x.sin 4x.tan 8x)) ?

$$\:\:\:\:\:\:\:\:\:\overset{\bigstar} {{b}}\overset{\bigstar} {{o}}\overset{\bigstar} {{b}}\overset{\Box} {{h}}\overset{\Box} {{a}}\overset{\Box} {{n}}\overset{\spadesuit} {{s}} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{tan}\:\mathrm{8}{x}−\mathrm{sin}\:\mathrm{4}{x}.\mathrm{cos}\:\mathrm{4}{x}}{{x}.\mathrm{sin}\:\mathrm{4}{x}.\mathrm{tan}\:\mathrm{8}{x}}\:? \\ $$

Question Number 108990 Answers: 2 Comments: 0