Straight Line and Pair of Straight Lines 1 Question 59

59. A line cuts the X-axis at A(7,0) and the Y-axis at B(0,5). A variable line PQ is drawn perpendicular to AB cutting the X-axis in P and the Y-axis in Q. If AQ and BP inters at R, find the locus of R.

(1990, 4M)

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Solution:

  1. The equation of the line AB is

x7+y5=1

5x7y=35

Equation of line perpendicular to AB is

7x+5y=λ

It meets X-axis at P(λ/7,0) and Y-axis at Q(0,λ/5).

The equations of lines AQ and BP are x7+5yλ=1 and 7xλy5=1, respectively.

Let R(h,k) be their point of intersection of lines AQ and BP.

Then, h7+5kλ=1

and 7hλk5=1

15k1h7=17h1+k5 [on eliminating λ ]

h(7h)=k(5+k)

h2+k27h+5k=0

Hence, the locus of a point is

x2+y27x+5y=0



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