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Question

From a point P perpendiculars PM and PN are drawn upon two fixed lines which are inclined at an angle ω, and which are taken as the axes of coordinates and meet in O; find the locus of P if MN pass through the fixed point (a,b).

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Solution

Let the point P be (h,k)

Draw PL parallel to MO and PL parallel to NO

In PLM,cosω=LMPL

OM=OL+LMOM=h+kcosω

So, the coordinates of M are (h+kcosω,0)

In PLN,cosω=LNPL

LN=PLcosω=hcosωON=OL+LNON=k+hcosω

So the coordinates of N are (0,k+hcosω)

Equation of MN is

y0=k+hcosω00(h+kcosω)(xhkcosω)y=k+hcosωh+kcosω(xhkcosω)yhykcosω=kxkhk2cosω+hxcosωh2cosωhkcos2ωh2cosω+k2cosω+hk(1+cos2ω)=h(y+xcosω)+k(x+ycosω)

It passes through (a,b)

h2cosω+k2cosω+hk(1+cos2ω)=h(b+acosω)+k(a+bcosω)

Replacing h by x and y by k

x2cosω+y2cosω+xy(1+cos2ω)=x(b+acosω)+y(a+bcosω)

is the required locus of P


696007_640698_ans_c04cf598690e4edbb1cdb6a464c36fe9.png

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