Find the value of `(x_(1)+x_(3)+x_(4)-x_(2))` for the equation `(3+x_(1)+x_(2)+x_(3)+x_(4))^(5)=9375x_(1)x_(2)x_(3)x_(4)`, where `x_(1),x_(2),x_(3),x_(4)gt0`

If the circle x^(2)+y^(2)=a^(2) intersects the hyperbola xy=c^(2) in four points P (x_(1) ,y_(1)) Q (x_(2), y_(2)) R (x_(3) ,y_(3)) S (x_(4) ,y_(4)) then 1) x_(1)+x_(2)+x_(3)+x_(4)=2c^(2) 2) y_(1)+y_(2)+y_(3)+y_(4)=0 3) x_(1)x_(2)x_(3)x_(4)=2c^(4) 4) y_(1)y_(2)y_(3)y_(4)=2c^(4)

If the mean of x_(1) and x_(2), and that of x_(1),x_(2),x_(3),x_(4) is M_(2), the mean of x_(1)+(x_(2))/(a),x_(3)+a,x_(4)-a is

The number of integral solution of x_(1)+x_(2)+x_(3)+x_(4)=25. Such that x_(1)>=1,x_(2)>=2,x_(3)>=3,x_(4)>=4

If the normal at the point P(x_(i),y_(i)),i=1,2,3,4 on the hyperbola xy=c^(2) are concurrent at the point Q(h,k) then _((x_(1)+x_(2)+x_(3)+x_(4))(y_(1)+y_(2)+y_(3)+y_(4)))((x_(1)+x_(2)+x_(3)+x_(4))(y_(1)+y_(2)+y_(3)+y_(4)))/(x_(1)x_(2)x_(3)x_(4)) is

Find the minimum value of log_(x_(1))(x_(2)-(1)/(4))+log_(x_(2))(x_(3)-(1)/(4))+......+log_(x_(20))(x_(1)-(1)/(4)) over all x_(1),x_(2),...x_(20)in((1)/(4),1)

Consider the system of equations x_1+x_(2)^(2)+x_(3)^(3)+x_(4)^(4)+x_(5)^(5)=5 and x_1+2x_2+3x_3+4x_4+5x_5=15 where x_1,x_2,x_3,x_4,x_5 are positive real numbers. Then numbers of (x_1,x_2,x_3,x_4,x_5) is ___________.

Let x_(1),x_(2), be the roots of the quadratic equation x^(2)+ax+b=0 and x_(3),x_(4), be the roots of the quadratic equation x^(2)-ax+b-2=0. If (1)/(x_(1))+(1)/(x_(2))+(1)/(x_(3))+(1)/(x_(4))=(5)/(6) and x_(1)x_(2)x_(3)x_(4)=24 then find the value of a .

If x_(1),x_(2),x_(3) and x_(4) are positive real numbers such that x_(1)x_(2)x_(3)x_(4)=1, then the least value of (1+x_(1))(1+x_(2))(1+x_(3))(1+x_(4)) is