Prove that: `|(a,b, ax+by),(b,c,bx+cy), (ax+by, bx+cy,0)|=(b^2-a c)(a x^2+2b x y+c y^2)` .

Answer

Proved

Explanation

\begin{aligned}
&\left|\begin{array}{ccc}
a & b & a x+b y \\
b & c & b x+c y \\
a x+b y & b x+c y & 0
\end{array}\right|\\ \\
&\text { Expanding along column (3). }\\ \\
&=(a x+b y)\left[b^{2} x+b y y-c a x-b p y\right]\\
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