If the system of equations $$2x+3y$$−$$z=0$$,$$x+ky$$−$$2z=3$$ and $$2x$$−$$y+z=0$$ has a $$non-trivial$$ solution x,y,zthen $$xy+yz+zx+k$$ is equal to

Question

If the system of equations  $$2x+3y$$−$$z=0$$,$$x+ky$$−$$2z=3$$ and $$2x$$−$$y+z=0$$  has a $$non-trivial$$ solution  x,y,zthen $$xy+yz+zx+k$$  is equal to

Infinity Learn · Accepted Answer

Given system of linear equations $$2x+3y$$−$$z=0x+ky$$−$$2z=0$$ And $$2x$$−$$y+z=0$$ has a $$non-trivial$$ solution (x$$,y,$$z)∴Δ$$=0$$⇒$$23$$−$$11k$$−$$22$$−$$11=02(k$$−$$2)$$−$$3(1+4)$$−$$1(1$$−$$2k)=0$$⇒$$2K$$−$$4$$−$$15+1+2K=0$$⇒ $$4K=18$$⇒$$K=92So$$, system of linear equations is $$2x+3y$$−$$z=0-----12x+9y$$−$$4z=0----2$$ And $$2x$$−$$y+z=0-----3From$$ Eqs. (i) and (ii) , we get $$6y$$−$$3z=0$$,$$yz=12From$$ Eqs. (i) and (iii), we get $$4x+2y=0$$⇒$$xy=$$−$$12so$$,$$xz=xy$$×$$yz=$$−$$14$$⇒$$zx=$$−$$4$$                    ∵  $$yz=12$$   and  $$xy=$$−$$12$$∴ $$xy+yz+zx+k=$$−$$12+12$$−$$4+92=12$$

If the system of equations $2 x + 3 y - z = 0, x + k y - 2 z = 3$ and $2 x - y + z = 0$ has a non-trivial solution $(x, y, z)$ then $\frac{x}{y} + \frac{y}{z} + \frac{z}{x} + k$ is equal to

Remember concepts with our Masterclasses.

Ready to Test Your Skills?

free study material

If the system of equations 2x+3y−z=0,x+ky−2z=3 and 2x−y+z=0 has a non-trivial solution x,y,zthen xy+yz+zx+k is equal to

Remember concepts with our Masterclasses.

Ready to Test Your Skills?

free study material

If the system of equations $2 x + 3 y - z = 0, x + k y - 2 z = 3$ and $2 x - y + z = 0$ has a non-trivial solution $(x, y, z)$ then $\frac{x}{y} + \frac{y}{z} + \frac{z}{x} + k$ is equal to