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Area of Polygon Using Coordinates

If Y = SX , Z...

Question

If Y = SX, Z = tX all the variables being differentiable functions of x and lower suffices denote the derivative with respect to x and $∣ ∣ ∣ ∣ \begin{matrix} X & Y & Z X_{1} & Y_{1} & Z_{1} X_{2} & Y_{2} & Z_{2} \end{matrix} ∣ ∣ ∣ ∣ \div ∣ ∣ ∣ \begin{matrix} S_{1} & t_{1} S_{2} & t_{2} \end{matrix} ∣ ∣ ∣ = X^{n}, t h e n n =$

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Solution

The correct option is C 3
$Δ = ∣ ∣ ∣ ∣ \begin{matrix} X & S X & t X X_{1} & S_{1} X + S_{1} X & t X_{1} + t_{1} X X_{2} & S X_{2} + 2 S_{1} X_{1} + S_{2} X & t X_{2} + 2 t_{1} X_{1} + t_{2} X \end{matrix} ∣ ∣ ∣ ∣ (\begin{matrix} C_{2} \to C_{2} - S C_{1} C_{3} \to C_{3} - S C_{1} \end{matrix}) = Δ = ∣ ∣ ∣ ∣ \begin{matrix} X & 0 & 0 X_{1} & S X_{1} & t_{1} X X_{2} & 2 S_{1} X_{1} + S_{2} X & 2 t_{1} X_{1} + t_{2} X \end{matrix} ∣ ∣ ∣ ∣ = X^{2} ∣ ∣ ∣ \begin{matrix} S_{1} & t_{1} 2 S_{1} X_{1} + S_{2} X & 2 t_{1} X_{1} + t_{2} X \end{matrix} ∣ ∣ ∣ = X^{3} ∣ ∣ ∣ \begin{matrix} S_{1} & t_{1} S_{2} & t_{1} \end{matrix} ∣ ∣ ∣ ∴ n = 3.$

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Y = S X, Z = T X

∣ ∣ ∣ ∣ \begin{matrix} X & Y & Z X_{1} & Y_{1} & Z_{1} X_{2} & Y_{2} & Z_{2} \end{matrix} ∣ ∣ ∣ ∣ = ∣ ∣ ∣ \begin{matrix} S_{1} & T_{1} S_{2} & T_{2} \end{matrix} ∣ ∣ ∣

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∣ ∣ ∣ ∣ \begin{matrix} x & y & z x_{1} & y_{1} & z_{1} x_{2} & y_{2} & z_{2} \end{matrix} ∣ ∣ ∣ ∣ = k (x) ∣ ∣ ∣ \begin{matrix} r_{1} & t_{1} r_{2} & t_{2} \end{matrix} ∣ ∣ ∣

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△ = ∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & x & x^{2} 1 & y & y^{2} 1 & z & z^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣

△_{1} = ∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 1 y z & z x & x y x & y & z \end{matrix} ∣ ∣ ∣ ∣,

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∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 1 x^{2} & y^{2} & z^{2} x^{3} & y^{3} & z^{3} \end{matrix} ∣ ∣ ∣ ∣ ∣

(x - y) (y - z) (z - x) (x y + y z + z x)

l_{1}, m_{1}, n_{1}; l_{2}, m_{2}, n_{2}; l_{3}, m_{3}, n_{3}

∣ ∣ ∣ ∣ \begin{matrix} l_{1} & m_{1} & n_{1} l_{2} & m_{2} & n_{2} l_{3} & m_{3} & n_{3} \end{matrix} ∣ ∣ ∣ ∣ = 0

\frac{x - x_{1}}{l_{1}} = \frac{y - y_{1}}{m_{1}} = \frac{z - z_{1}}{n_{1}}, \frac{x - x_{2}}{l_{2}} = \frac{y - y_{2}}{m_{2}} = \frac{z - z_{2}}{n_{2}}

∣ ∣ ∣ ∣ \begin{matrix} x - x_{1} & y - y_{1} & z - z_{1} l_{1} & m_{1} & n_{1} l_{2} & m_{2} & n_{2} \end{matrix} ∣ ∣ ∣ ∣ = 0

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