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Trong tam giác vuông ABP:
\(tanP=\dfrac{AB}{AP}\Rightarrow AP=\dfrac{AB}{tanP}\Rightarrow PQ+AQ=\dfrac{AB}{tanP}\) (1)
Trong tam giác vuông ABQ:
\(tanQ=\dfrac{AB}{AQ}\Rightarrow AQ=\dfrac{AB}{tanQ}\) (2)
\(\left(1\right);\left(2\right)\Rightarrow PQ+\dfrac{AB}{tanQ}=\dfrac{AB}{tanP}\Rightarrow PQ=AB\left(\dfrac{1}{tanP}-\dfrac{1}{tanQ}\right)\)
\(\Rightarrow AB=\dfrac{PQ}{\dfrac{1}{tanP}-\dfrac{1}{tanQ}}=\dfrac{100}{\dfrac{1}{tan15^0}-\dfrac{1}{tan55^0}}\approx33\left(m\right)\)
\(A=4\left[\left(sin^2a+cos^2a\right)^3-3sin^2a.cos^2a\left(sin^2a+cos^2a\right)\right]-6\left[\left(sin^2a+cos^2a\right)^2-2sin^2a.cos^2a\right]\)
\(=4\left(1-3sin^2a.cos^2a\right)-6\left(1-2sin^2a.cos^2a\right)\)
\(=4-12sin^2a.cos^2a-6+12sin^2a.cos^2a\)
\(=-2\)
\(\dfrac{-1}{39}+\dfrac{-1}{52}=\dfrac{-7}{156}\)
\(\dfrac{-6}{9}+\dfrac{-12}{16}=\dfrac{-17}{12}\)
\(\dfrac{-2}{5}-\dfrac{-3}{11}=\dfrac{-7}{55}\)
\(\dfrac{-34}{37}.\dfrac{74}{-85}=\dfrac{4}{5}\)
\(\dfrac{-5}{9}:\dfrac{-7}{18}=\dfrac{10}{7}\)
Chúc bạn học tốt!!!
a) \(\left(-\dfrac{1}{39}\right)+\left(-\dfrac{1}{52}\right)=\dfrac{-4-3}{156}=-\dfrac{7}{156}\)
b) \(\left(-\dfrac{6}{9}\right)+\left(-\dfrac{12}{16}\right)=-\dfrac{6}{9}-\dfrac{12}{16}=-\dfrac{17}{12}\)
c) \(-\dfrac{2}{5}-\left(-\dfrac{3}{11}\right)=-\dfrac{2}{5}+\dfrac{3}{11}=-\dfrac{7}{55}\)
d) \(\left(-\dfrac{34}{37}\right)\cdot\left(-\dfrac{74}{85}\right)=2\cdot\dfrac{2}{5}=\dfrac{4}{5}\)
e) \(\left(-\dfrac{5}{9}\right):\left(-\dfrac{7}{18}\right)=\dfrac{5}{9}\cdot\dfrac{18}{7}=5\cdot\dfrac{2}{7}=\dfrac{10}{7}\)
37:
\(AB=\sqrt{\left(-2-1\right)^2+\left(4-2\right)^2}=\sqrt{13}\)
\(AC=\sqrt{\left(3-1\right)^2+\left(5-2\right)^2}=\sqrt{13}\)
\(BC=\sqrt{\left(3+2\right)^2+\left(5-4\right)^2}=\sqrt{26}\)
Vì AB^2+AC^2=BC^2 và AB=AC
nên ΔABC vuông cân tại A
=>S ABC=1/2*AB*AC=1/2*13=13/2
AH=13/2*2:căn 26=13/căn 26=1/2*căn 26
36.
\(Q=x-2+\dfrac{3}{x-2}+2\ge2\sqrt{\dfrac{3\left(x-2\right)}{x-2}}+2=2\left(\sqrt{3}+1\right)\)
39.
\(\dfrac{\sqrt{3}cosx+sinx}{2cosx+3sinx}=\dfrac{\dfrac{\sqrt{3}cosx}{sinx}+\dfrac{sinx}{sinx}}{\dfrac{2cosx}{sinx}+\dfrac{3sinx}{sinx}}=\dfrac{\sqrt{3}cotx+1}{2cotx+3}=\dfrac{\sqrt{3}.\left(-\dfrac{1}{2}\right)+1}{2.\left(-\dfrac{1}{2}\right)+3}=\dfrac{2-\sqrt{3}}{4}\)
\(\Rightarrow\left\{{}\begin{matrix}a=2\\b=1\end{matrix}\right.\) \(\Rightarrow a-4b=-2\)
40. \(\overrightarrow{BA}=\left(6;4\right)=2\left(3;2\right)\Rightarrow\) trung trực AB nhận (3;2) là 1 vtpt
Gọi M là trung điểm AB \(\Rightarrow M\left(-1;3\right)\)
Phương trình trung trực AB:
\(3\left(x+1\right)+2\left(y-3\right)=0\Leftrightarrow3x+2y-3=0\)
Đường tròn (C) tâm \(I\left(1;-2\right)\) bán kính \(R=3\)
\(S_{IAB}=\dfrac{1}{2}IA.IB.sin\widehat{AIB}=\dfrac{1}{2}R^2.sin\widehat{AIB}\le\dfrac{1}{2}R^2\)
\(S_{max}\) khi \(sin\widehat{AIB}=1\Rightarrow\Delta AIB\) vuông cân tại I
\(\Rightarrow AB=R\sqrt{2}=3\sqrt{2}\)
\(\Rightarrow d\left(I;AB\right)=\dfrac{AB}{2}=\dfrac{3\sqrt{2}}{2}\)
Gọi phương trình AB có dạng: \(a\left(x+1\right)+b\left(y+3\right)=0\) với a;b ko đồng thời bằng 0
\(d\left(I;AB\right)=\dfrac{\left|a-2b+a+3b\right|}{\sqrt{a^2+b^2}}=\dfrac{3\sqrt{2}}{2}\)
\(\Leftrightarrow\sqrt{2}\left|2a+b\right|=3\sqrt{a^2+b^2}\)
\(\Leftrightarrow2\left(4a^2+4ab+b^2\right)=9\left(a^2+b^2\right)\)
\(\Leftrightarrow a^2-8ab+7b^2=0\Rightarrow\left[{}\begin{matrix}a=b\\a=7b\end{matrix}\right.\)
Chọn b=1 \(\Rightarrow\left[{}\begin{matrix}\left(a;b\right)=\left(1;1\right)\\\left(a;b\right)=\left(1;7\right)\end{matrix}\right.\)
Có 2 đường thẳng thỏa mãn: \(\left[{}\begin{matrix}1\left(x+1\right)+1\left(y+3\right)=0\\1\left(x+1\right)+7\left(y+3\right)=0\end{matrix}\right.\)