Vertices of a variable triangle are (3 , 4) $(5\cos \theta ,5\sin \theta )$  and $(5\sin \theta ,-5\cos \theta )$ . Then locus of its orthocenter is                                  

Circumcentre of the triangle is (0 , 0) and  Centroid $\left(\frac{3+5\cos \theta +5\sin \theta }{3},\frac{4+5\sin \theta -5\cos \theta }{3}\right)$                                                                     [$\because$  Centroid divides the join of orthocenter and circumentre in 2 : 1 ratio $\Rightarrow G=\frac{2S+O}{3} If S=(0,0) then 3G=O$

let (h , k) represents orthocenter and  $3G=3+5\cos \theta +5\sin \theta ,4+5\sin \theta -5\cos \theta$  

$\therefore h=3+5\cos \theta +5\sin \theta$           $k=4+5\sin \theta -5\cos \theta$   

Adding and subtracting the above equations we get                           

$\Rightarrow \sin \theta =\frac{h+k-7}{10};\cos \theta =\frac{h-k+1}{10}$ 

squarring and adding                                                                                                            

$\Rightarrow {(h+k-7)}^2+{(h-k+1)}^2=100$                                                                                      

 $\therefore$  Locus of orthocenter is   ${(x+y-7)}^2+{(x-y+1)}^2=100$