lf $x_{1}, x_{2}, x_{3}$ are the abscissa of the points, $A_{1}, A_{2}, A_{3}$ respectively where the lines $y = m_{1} x, y = m_{2} x$ and $y = m_{3} x$ meet the line $2 x - y + 3 = 0$ such that $m_{1}, m_{2}, m_{3}$ are in $A . P .$ then $x_{1}, x_{2}, x_{3}$ are in

A.P.

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G.P.

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H.P.

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A.G.P.

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Solution

The correct option is D H.P.
Let, $y = m x$ is a line which meets $2 x - y + 3 = 0$
Subsitute $y = m x$ in $2 x - y + 3 = 0$ to get
$x = \frac{3}{m - 2}$ & $y = \frac{3 m}{m - 2}$
$\Rightarrow∴ x_{1} = \frac{3}{m_{1} - 2}$ & $y_{1} = \frac{3 m_{1}}{m_{1} - 2}$
$x_{2} = \frac{3}{m_{2} - 2}, x_{3} = \frac{3}{m_{3} - 2}$

Given, $m_{1}, m_{2}, m_{3}$ are in A. P.
So, $m_{1} - 2, m_{2} - 2, m_{3} - 2$ are also in A.P.
$\frac{1}{m_{1} - 2}, \frac{1}{m_{2} - 2}, \frac{1}{m_{3} - 2}$ are in H.P.
$\frac{3}{m_{1} - 2}, \frac{3}{m_{2} - 2}, \frac{3}{m_{3} - 2}$ are also in H.P.

$\Rightarrow x_{1}, x_{2}, x_{3}$ are in H.P.

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Similar questions

Q. Find the conditions that the straight lines

y = m_{1} x + a_{1}, y = m_{2} x + a_{2}

, and

y = m_{3} x + a_{3}

may meet in a point.

Q. Find the area of the triangle formed by the straight lines whose equations are

y = m_{1} x + \frac{a}{m_{1}}

y = m_{2} x + \frac{a}{m_{2}}

, and

y = m_{3} x + \frac{a}{m_{3}}

Show that the are of the triangle formed by the lines $y = m_{1} x, y = m_{2}$ x and y = c is equal to $\frac{c^{2}}{4} (\sqrt{33} + \sqrt{11})$ , where $m_{1}, m_{2}$ are the roots of the equation $x^{2} + (\sqrt{3} + 2) x + \sqrt{3} - 1 = 0.$

Q. If

m_{1}

and

m_{2}

are the roots of the equation

x^{2} + (\sqrt{3} + 2) x + (\sqrt{3} - 1) = 0

, then the area of the triangle formed by the lines

y = m_{1} x, y = m_{2} x

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is :

Q. Show that the area of the triangle formed by the lines y = m₁ x, y = m₂ x and y = c is equal to

\frac{c^{2}}{4} (\sqrt{33} + \sqrt{11}),

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lf x1, x2, x3 are the abscissa of the points, A1, A2, A3 respectively where the lines y=m1x,y=m2x and y=m3x meet the line 2x−y+3=0 such that m1,m2,m3 are in A.P. then x1, x2, x3 are in

lf $x_{1}, x_{2}, x_{3}$ are the abscissa of the points, $A_{1}, A_{2}, A_{3}$ respectively where the lines $y = m_{1} x, y = m_{2} x$ and $y = m_{3} x$ meet the line $2 x - y + 3 = 0$ such that $m_{1}, m_{2}, m_{3}$ are in $A . P .$ then $x_{1}, x_{2}, x_{3}$ are in