If p≠a,q≠b,r≠cand the system of equations $$px+ay+az=0$$,$$bx+qy+bz=0$$,$$cx+cy+rz=0$$ has a $$non-trivial$$ solution, then the value of pp−$$a+qq$$−$$b+rr$$−$$c=_________$$

Question

If p≠a,q≠b,r≠cand the system of equations  $$px+ay+az=0$$,$$bx+qy+bz=0$$,$$cx+cy+rz=0$$ has a $$non-trivial$$ solution, then the value of pp−$$a+qq$$−$$b+rr$$−$$c=_________$$

Infinity Learn · Accepted Answer

The condition for a homogeneous system to have a $$non-trivial$$ solution is Δ$$=0$$⇒p   a   ab   q     bc    c     $$r=0$$   $$c3$$→$$c3$$−$$c1$$ &$$c2$$→$$c2$$−$$c1$$⇒p            a−p          a−pb            q−b               $$0c$$                 $$0$$                        r−$$c=0Expanding$$ along $$C3we$$ $$get(a-p)bq-bc0+(r-c)p-pb=0q-b=0(a-p)$$[$$-c(q-b)$$]$$+(r-c)$$[$$p(q-b)-b(a-p)$$]$$=0p$$−aq−$$bc+r$$−cq−$$bp+br$$−cp−$$a=0Dividing$$ with p−aq−br−c we getcr−$$c+pp$$−$$a+bq$$−$$b=0$$⇒pp−$$a+qq$$−$$b+rr$$−$$c=2$$

If $p \neq a, q \neq b, r \neq c$ and the system of equations $p x + a y + a z = 0, b x + q y + b z = 0, c x + c y + r z = 0$ has a non-trivial solution, then the value of $\frac{p}{p - a} + \frac{q}{q - b} + \frac{r}{r - c} =_________$

Remember concepts with our Masterclasses.

Ready to Test Your Skills?

free study material

If p≠a,q≠b,r≠cand the system of equations px+ay+az=0,bx+qy+bz=0,cx+cy+rz=0 has a non-trivial solution, then the value of pp−a+qq−b+rr−c=_________

Remember concepts with our Masterclasses.

Ready to Test Your Skills?

free study material

If $p \neq a, q \neq b, r \neq c$ and the system of equations $p x + a y + a z = 0, b x + q y + b z = 0, c x + c y + r z = 0$ has a non-trivial solution, then the value of $\frac{p}{p - a} + \frac{q}{q - b} + \frac{r}{r - c} =_________$