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  • Fill in the blanks Locus of the mid-points of the portion of the line x sin θ + y cos θ = p intercepted between the axes is ____.

Fill in the blanks

Locus of the mid-points of the portion of the line x sin θ + y cos θ = p intercepted between the axes is ____.

Answers (1)

Given equation of the line is x sinθ+y cosθ=p…..(i)

 Let Ph,kbe the midpoint of the given line where it meets the two axis at a,0and 0,b.

 Since (a,0) lies on equation (i) then asinθ+0=p   a= \frac{p}{\sin \theta }…..(ii)

  (0,b) also lies on the equation i then 0+bcosθ=p

 b=\frac{p}{\cos \theta }……(iii) 

Since, P (h,k) is the midpoint of the given line  h=\frac{a+0}{2}=\frac{a}{2}

  2h=a   and k=\frac{0+b}{2}=\frac{b}{2}

  2k=b 

  Putting the value of a=2h in equation (ii) we get  2h=\frac{p}{\sin \theta }

\sin \theta =\frac{p}{2k }…..(iv) 

Putting the value of b=2k in equation (ii) we get 2k =\frac{p}{\cos \theta }

\cos \theta =\frac{p}{2k }….(v)

 Squaring and adding equation (iv) and (v), we get \sin^{2}\theta + \cos^{2}\theta=\left ( \frac{p}{2h} \right )^{2}+\left ( \frac{p}{2k} \right )^{2}

 1= \frac{p^{2}}{4h^{2}}+ \frac{p^{2}}{4k^{2}}    or  1= \frac{p^{2}}{4x^{2}}+ \frac{p^{2}}{4y^{2}}

 or  4x2y2=p2y2+p2x2 

  or 4x2y2=p2(x2+y2) is the locus of the mid-points of the portion of the line intercepted between the axes

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