Through a fixed point any straight line is drawn meeting two given parallel straight lines in P and Q, through P and Q straight lines are drawn in fixed directions, meeting in R. Prove that the locus of R is straight line.
Text Solution
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Take the fixed point O for origin, and the axis of y parallel to the two parallel straight lines, and let the equations of these parallel lines be x = a, x = b.
Then, if the equation of OPQ be y = mx, the abscissa of Q is b, and therefore its ordinate mb. Let PR
be always parallel to y = m'x and QR always parallel to y = m"x, then the equation of PR, QR will be
y – ma = m'(x – a) .......(i)
y – mb = m"(x – b) .........(ii)
To find locus we have only to eliminate m between the equation (i) and (ii)
The result is (b – a) y = m'b(x – a) – m" a (x – b).
This equation is of the first degree, and therefore the required locus is a straight line.
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