question_answer

A triangle ABC satisfies the relation \[2\sec 4C+{{\sin }^{2}}2A+\sqrt{\sin B}=0\]and a point P is taken on the longest side of the triangle such that it divides the side in the ratio 1:3. Let Q and R be the circum centre and orthocenter of\[\Delta ABC\]. If PQ: QR; RP= 1 : \[\alpha :\beta ,\], then the value of \[{{\alpha }^{2}}+{{\beta }^{2}}.\]

A)  9                                            

B)  8

C)  6                                            

D)  7

Correct Answer: A

Solution :

 \[2\sec 4C+{{\sin }^{2}}2A+\sqrt{\sin B}=0\] \[A={{45}^{o}},B={{90}^{o}}\]and\[C={{45}^{o}}\] Let\[AQ=a,\]then\[BP=\frac{a}{2},\]\[PQ=\frac{a}{2}\]and\[QR=a\] \[\therefore \]\[PR=\sqrt{{{a}^{2}}+\frac{{{a}^{2}}}{4}}=\frac{\sqrt{5a}}{2}\] \[\therefore \]\[1:\alpha :\beta =\frac{a}{2}:a:\frac{\sqrt{5a}}{2}=1:2:\sqrt{5}\] \[\therefore \]\[\alpha =2\]\[\beta =\sqrt{5}\] \[\therefore \]\[{{\alpha }^{2}}+{{\beta }^{2}}=9\]