X-y - x - x 15 x-4 y - 4 x - 2 x 10 x-8 y - 8 x - 3 xend vmatrix - x - 3

LHS: Let Δ = x+y amp; x amp;x 5x+4y amp;4x amp;2x 10x+8y amp;8x amp;3x [Applying R_1→ R_2 4R_1 and R_3→ R_3 8R_1 = x+y amp; ...

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Summation of Determinant

x+y & x & x 1...

Question

Using properties of determinants, prove that: $∣ ∣ ∣ ∣ \begin{matrix} x + y & x & x 5 x + 4 y & 4 x & 2 x 10 x + 8 y & 8 x & 3 x \end{matrix} ∣ ∣ ∣ ∣ = x^{3}$

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∣ ∣ ∣ ∣ \begin{matrix} x + y & x & x 5 x + 4 y & 4 x & 2 x 10 x + 8 y & 8 x & 3 x \end{matrix} ∣ ∣ ∣ ∣ = x^{3}

⎡ ⎢ ⎣ \begin{matrix} x + y & x & x 5 x + 4 y & 4 x & 2 x 10 x + 8 y & 8 x & 3 x \end{matrix} ⎤ ⎥ ⎦ = x^{3}

y^{2} - 4 x - 8 y + 40 = 0

x^{2} - 8 x - 4 y + 40 = 0

y^{2} = 4 x

x y = 16

x^{2} + y^{2} - 4 x - 6 y - 3 = 0, x^{2} + y^{2} + 8 x - 4 y + 11 = 0

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