71Now that's what I call a solution!
1) If the curves ax2 + 4xy + 2y2 + x + y + 5 = 0 and ax2 + 6xy + 5y2 + 2x + 3y + 8 = 0 intersect at four concyclic points, then the value of |a| is...
2) For 0 < x < pi/2 the solution of \sum_{m=1}^{6}{} cosec {x + (m - 1)pi/4} cosec (x + mpi/4) = 4√2 is/are
(A) pi/4
(B) pi/6
(C) pi/12
(D) 5pi/12
3) Tangents are drawn to 3x2 - 2y2 = 6 from a point P. If these tangents intersect the coordinate axes at concylic points then the locus of P is
(A) x2 + y2 = 5
(B) x2 - y2 = 5
(C) 1/x2 + 1/y2 = 1/5
(D) none of these
Two more questions regarding rectangular hyperbola has been started as a fresh thread...!!
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7 Answers
2623) use condition of tangency.
let the concyclic points be (a,0) (b,0) (0,c) (0,d)
then ab=cd
Kashish Singh full solution plzzz
Upvote·0· Reply ·2015-11-20 15:38:02
2621) let the angle between the pair of lines in first eqn. be θ
then the angle between the pair of lines in 2nd eqn. = (180 -θ)
also use tanθ = 2√h2-aba+b
1708\hspace{-16}\bf{\sum_{m=1}^{6}\csc\left(x+(m-1)\frac{\pi}{4}\right).\csc\left(x+m\frac{\pi}{4}\right)=4\sqrt{2}}$\\\\\\ $\bf{\sum_{m=1}^{6}\frac{1}{\sin\left(x+(m-1)\frac{\pi}{4}\right).\sin\left(x+m\frac{\pi}{4}\right)}=4\sqrt{2}}$\\\\\\ $\bf{\frac{1}{\sin \left(\frac{\pi}{4}\right)}.\sum_{m=1}^{6}\frac{\sin \left\{(x+m.\frac{\pi}{4})-(x+(m-1).\frac{\pi}{4})\right\}}{\sin\left(x+(m-1)\frac{\pi}{4}\right).\sin\left(x+m\frac{\pi}{4}\right)}=4\sqrt{2}}$\\\\\\ $\bf{\sum_{m=1}^{6} \cot (x+(m-1).\frac{\pi}{4})-\cot (x+m.\frac{\pi}{4})=4}$\\\\\\ Now Using Telescopic Sum, We Get\\\\\\ $\bf{\cot (x)-\cot\left(x+\frac{6\pi}{4}\right)=4}$
71@Aditya.
How can we say that those two eqautions are for a pair of straight lines?
262ohhh ! sorry my mistake.
btw there's a much easier way to do the sum.
let first eqn be S=0
and 2nd eqn. be P=0
thus eqn of any curve passing through their intersection is
S+kP =0
since the points are concyclic, the above curve should be a circle , thus
coeff. of x2 = coeff. of y2
and coeff. of xy =0
applying these conditions we get a= -4
or |a| =4