The value of $\begin{vmatrix}a^2&-ab&-bc\\-ab&b^2&-bc\\ca&bc&-c^2\end{vmatrix}$, is

$4a^2\, b^2\,c^2$

We have,

$Δ=\begin{vmatrix}a^2&-ab&-bc\\-ab&b^2&-bc\\ca&bc&-c^2\end{vmatrix}$

$⇒Δ=abc\begin{vmatrix}a&-b&-c\\-a&b&-c\\a&b&-c\end{vmatrix}$ Taking a, b, c common from $R_1, R_2$ and $R_3$ respectively

$⇒Δ=a^2b^2c^2\begin{vmatrix}1&-1&-1\\-1&1&-1\\1&1&-1\end{vmatrix}$ Taking a, b, c common from $C_1, C_2$ and $C_3$ respectively

$⇒Δ=a^2b^2c^2\begin{vmatrix}1&0&0\\-1&0&-2\\1&2&0\end{vmatrix}$ Applying $C_2 →C_2 + C_1, C_3 → C_3 +C_1$

$⇒Δ=4a^2\, b^2\,c^2$