Question

A line cuts x-axis at A(7, 0) and y-axis at B(0, 5). A variable line PQ is drawn perpendicular to AB cutting the x-axis at P and the y-axis at Q. If AQ and BP intersect at R, then find the locus of R.

Answer 1

See Fig.  The slope of  (bar)AB is

-5 - 0/0 - 7 = 5/7

Therefore, the slope of PQ is –7/5. Consider ΔABQ in which QP is the altitude from Q onto AB and AP is the altitude from A onto BQ. These two intersect at P. Hence, BP is the third altitude of  ΔABQ. Therefore, BR is perpendicular to AR. Hence, if R = (h, k), then

Slope of  BR  x  Slope of AR = -1

⇒ (k + 5/h)(k/h -7) = -1

⇒ h² + k² - 7h + 5k = 0

Therefore, the locus of R(h, k) is

x² + y² - 7x + 5y = 0