Let R be a point on a variable line passing through origin and which cuts the lines 2x+ 5y-3 =0 and 4x+7y-3=0 at P,Q such that OP,OR,OQ are in harmonic progression. Then the locus of R

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Passes through (3,-1)

$Form a triangle of area \frac{3}{2} squnits with given lines.$

is a bounded curve

$Encloses an area of \frac{7 π}{2} Sq.units$

answer is A.

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Detailed Solution

$l e t R = (r \cos θ, r \sin θ), P = (r_{1} \cos θ, r_{1} \sin θ), Q = (r_{2} \cos θ, r_{2} \sin θ) (p a r a m e t r i c c o o r d i n a t e s)$

$P l i e s o n t h e l i n e 2 x + 5 y - 3 = 0 \Rightarrow 2 (r_{1} \cos θ) + 5 (r_{1} \sin θ) - 3 = 0 \Rightarrow 2 \cos θ + 5 \sin θ = \frac{3}{r_{1}}$

$a n d \sin c e Q l i e s o n t h e l i n e 4 x + 7 y - 3 = 0 \Rightarrow 4 \cos θ + 7 \sin θ = \frac{3}{r_{2}}$

$g i v e n t h a t r_{1,} r, r_{2} a r e i n H . P \Rightarrow$

$\begin{array}{l} \frac{2}{r} = \frac{1}{r_{1}} + \frac{1}{r_{2}} = \frac{2 \cos θ + 5 \sin θ}{3} + \frac{4 \cos θ + 7 \sin θ}{3} \\ 6 = 6 r \cos θ + 12 r \sin θ \Rightarrow \\ x + 2 y = 1 is the locus of R \end{array}$