Suppose a, b and c are distinct real numbers. Let Δ$$=aa+ca$$−bb−$$cba+bc+bc$$−$$ac=0$$ .Then the straight line $$a(x$$−$$5)+b(y$$−$$2)+c=0$$ passes through the fixed point

Question

Suppose a, b and c are distinct real numbers. Let Δ$$=aa+ca$$−bb−$$cba+bc+bc$$−$$ac=0$$ .Then the straight line $$a(x$$−$$5)+b(y$$−$$2)+c=0$$ passes through the fixed point

Infinity Learn · Accepted Answer

Applying $$C1$$→$$aC1+bC2+cC3$$ we getLet          Δ$$=1aa(a+b+c)a+ca$$−$$bb(a+b+c)ba+bc(a+b+c)c$$−$$ac=1aa(a+b+c)$$Δ$$1$$, whereΔ$$1=aa+ca$$−$$bbba+bcc$$−acUsing $$C2$$→$$C2$$−$$C1$$,$$C3$$→$$C3$$−$$C1$$ we getΔ$$1=ac$$−$$bb0ac$$−$$a0=aa2+b2+c2$$∴ Δ$$=(a+b+c)a2+b2+c2=0As$$ a, b, c are distinct real numbers, $$a2+b2+c2$$≠$$0$$ $$thereforea+b+c=0$$⇒the line $$a(x$$–$$5)+b(y$$–$$2)+c=0$$ passes through (6$$, $$3)

Suppose $a, b$ and $c$ are distinct real numbers. Let $Δ = |\begin{array}{ccc} a & a + c & a - b \\ b - c & b & a + b \\ c + b & c - a & c \end{array}| = 0$ .Then the straight line $a (x - 5) + b (y - 2) + c = 0$ passes through the fixed point

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Suppose a, b and c are distinct real numbers. Let Δ=aa+ca−bb−cba+bc+bc−ac=0 .Then the straight line a(x−5)+b(y−2)+c=0 passes through the fixed point

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Suppose $a, b$ and $c$ are distinct real numbers. Let $Δ = |\begin{array}{ccc} a & a + c & a - b \\ b - c & b & a + b \\ c + b & c - a & c \end{array}| = 0$ .Then the straight line $a (x - 5) + b (y - 2) + c = 0$ passes through the fixed point