Find the equation of the line joining (1, 2) | vedclass

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(A) Let $P(x, y)$ be any point on the line joining the points $A(1, 2)$ and $B(3, 6)$. Since the points $A, B,$ and $P$ are collinear,the area of the

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$y = 2x$

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(A) Let $P(x, y)$ be any point on the line joining the points $A(1, 2)$ and $B(3, 6)$. Since the points $A, B,$ and $P$ are collinear,the area of the triangle formed by these three points must be zero.The area of a triangle with vertices $(x_1, y_1), (x_2, y_2),$ and $(x_3, y_3)$ is given by $\frac{1}{2} |\det(M)| = 0$,where $M = \begin{bmatrix} x_1 & y_1 & 1 \ x_2 & y_2 & 1 \ x_3 & y_3 & 1 \end{bmatrix}$.Substituting the given points:$\frac{1}{2} \left| \begin{matrix} 1 & 2 & 1 \ 3 & 6 & 1 \ x & y & 1 \end{matrix} ight| = 0$Expanding the determinant along the first row:$1(6(1) - y(1)) - 2(3(1) - x(1)) + 1(3y - 6x) = 0$$(6 - y) - 2(3 - x) + (3y - 6x) = 0$$6 - y - 6 + 2x + 3y - 6x = 0$Combining like terms:$(2x - 6x) + (3y - y) + (6 - 6) = 0$$-4x + 2y = 0$$2y = 4x$$y = 2x$Thus,the equation of the line is $y = 2x$.

Find the equation of the line joining $(1, 2)$ and $(3, 6)$ using determinants.

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