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If =Xcostheta-Ysintheta,q=Xsintheta+Ycosthetaandp^(2)+4pq+q^(2)=AX^(2)+BY^(2),0le0le(pi)/(2).What is the value of A ? |
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Answer» 4 `Q=xsintheta+ycostheta` Given, `p^(2)+4pq+q^(2)=Ax^(2)+By^(2)` Letus take`theta=(pi)/(4)`. `p=X"COS"(pi)/(4)-y" cos"(pi)/(4)=(x+y)/(sqrt(2))` `q=x" SIN"(pi)/(4)+y" cos"(pi)/(4)=(x+y)/(sqrt(2))` `pq=(x^(2)-y^(2))/(2)rArr2pq=x^(2)-y^(2)rArr4pq=2x^(2)-2y^(2)` . . . (1) Now, `p^(2)+q^(2)=x^(2)cos^(2)theta+y^(2)sin^(2)theta-2xycosthetasintheta+x^(2)sin^(2)theta+y^(2)cos^(2)theta+2xysinthetacostheta=x^(2)+y^(2)`. . . (2) From(1) , (2) , `p^(2)+q^(2)+4pq=x^(2)+y^(2)+2x^(2)-2y^(2)=3x^(2)-y^(2)` Comparingthis with the given from , we get `theta=(pi)/(4),A=3,B=-1` |
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