Question

146. The line 2x + 3y = 6, 2x+3y=8 cut the x-axis at 4. B respectively. A line l drawn through
the point (2, 2) meets the x-axis at C in such a way that abscissae of A,B and C are in
arithmetic progression. Then the equation of the line l is
[EAMCET 2001]
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Answer 1

EXPLANATION.

$\sf : \implies \: line \: 2x + 3y = 6 \: \: and \: \: 2x + 3y = 8 \: \: cuts \: x \: axis \: at \: 4.b \\ \\ \sf : \implies \: line \: l \: is \: drawn \: through \: the \: point \: (2 , 2) \: meets \: the \: x \: axis \: at \: c. \\ \\ \sf : \implies \: abscissa \: of \: a,b \: \: and \: \: c \: \: are \: in \: ap$

$\sf : \implies \: equation \: of \: the \: line \: from \: point \: (2 , 2) \\ \\ \sf : \implies \: (y - y_{1}) = m(x - x_{1}) \\ \\ \sf : \implies \: (y - 2) = m(x - 2) \\ \\ \sf : \implies \: y - 2 = mx \: - 2m \\ \\ \sf : \implies \: y \: = mx + 2 - 2m$

$\sf : \implies \: at \: x \: axis \: y \: = 0 \\ \\ \sf : \implies \: (0) = mx \: + 2 - 2m \\ \\ \sf : \implies \: x \: = \frac{2(m - 1)}{m} \\ \\ \sf : \implies \: coordinates \: are \: ( \frac{2(m - 1)}{m} , \: 0)$

$\sf : \implies \: abscissae \: a,b,c \: \: are \: in \: ap \: (for \: x \: coordinates) \\ \\ \sf : \implies \: 3,4, \: \frac{2(m - 1)}{m} are \: in \: ap \: \\ \\ \sf : \implies \: conditions \: of \: an \: ap \: \implies \: 2b = a + c$

$\sf : \implies \: 2 \times 4 = 3 + \dfrac{2(m - 1)}{m} \\ \\ \sf : \implies \: 8 = 3 + \frac{2(m - 1)}{m} \\ \\ \sf : \implies \: 5 = \frac{2(m - 1)}{m} \\ \\ \sf : \implies \: 5m \: = 2m - 2 \\ \\ \sf : \implies \: m \: = - \frac{2}{3}$

$\sf : \implies \: equation \: of \: line \: \\ \\ \sf : \implies \: y \: = \frac{ - 2}{3}x \: \: + \: \: 2 \: \: + \: \frac{2 \times 2}{3} \\ \\ \sf : \implies \: y \: = \frac{ - 2x}{3} + 2 + \frac{4}{3} \\ \\ \sf : \implies \: y = \frac{ - 2x + 6 + 4}{3} \\ \\ \sf : \implies \: 3y = - 2x + 10 \\ \\ \sf : \implies \: 3y + 2x = 10$

$\sf : \implies \: \green{{ \underline{equation \: of \: line \: = 3y + 2x = 10}}}$