If the system of linear equations $$x+ky+3z=0$$,$$3x+ky$$−$$2z=0$$ $$2x+4y$$−$$3z=0$$ has a $$non-zero$$ solution x,y,z , then $$xzy2$$ is equal to

Question

If the system of linear equations  $$x+ky+3z=0$$,$$3x+ky$$−$$2z=0$$    $$2x+4y$$−$$3z=0$$  has a $$non-zero$$ solution  x,y,z  , then  $$xzy2$$ is equal to

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We have ,$$x+ky+3z=0$$;  $$3x+ky$$−$$2z=0$$;  $$2x+4y$$−$$3z=0System$$ of equation has $$non-zero$$ solution, if $$1k33k$$−$$224$$−$$3=0$$⇒$$($$−$$3k+8)$$−$$k($$−$$9+4)+3(12$$−$$2k)=0$$⇒ −$$3k+8+9k$$−$$4k+36$$−$$6k=0$$⇒−$$4k+44=0$$⇒$$k=11$$ Let  $$z=$$λ, then we get $$x+11y+3$$λ$$=0----13x+11y$$−$$2$$λ$$=0-----2$$ And  $$2x+4y$$−$$3$$λ$$=0----3Solving$$ Eqs. (i) and (ii), we get $$x=5$$λ$$2$$,$$y=$$−λ$$2$$,$$z=$$λ⇒$$xzy2=5$$λ$$22$$×−λ$$22=10$$

If the system of linear equations $x + k y + 3 z = 0, 3 x + k y - 2 z = 0 2 x + 4 y - 3 z = 0$ has a non-zero solution $(x, y, z)$ , then $\frac{x z}{y^{2}}$ is equal to

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If the system of linear equations x+ky+3z=0,3x+ky−2z=0 2x+4y−3z=0 has a non-zero solution x,y,z , then xzy2 is equal to

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If the system of linear equations $x + k y + 3 z = 0, 3 x + k y - 2 z = 0 2 x + 4 y - 3 z = 0$ has a non-zero solution $(x, y, z)$ , then $\frac{x z}{y^{2}}$ is equal to