question_answer

A triangle ABC satisfies the relation \[2\sec 4C\]\[+{{\sin }^{2}}\text{ }2A+\sqrt{\sin B}=0\] 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 circumcentre and orthocentre of A \[\Delta ABC.\]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\[QR=a\] \[\therefore \]\[PR=\sqrt{{{a}^{2}}+\frac{{{a}^{2}}}{4}}=\frac{\sqrt{5}a}{2}\] \[\therefore \]\[1:\alpha :\beta =\frac{a}{2}:a:\frac{\sqrt{5}a}{2}=1:2:\sqrt{5}\] \[\therefore \]\[\alpha =2\]and\[\beta =\sqrt{5}\]\[\therefore \]