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Straight lines are formed passing through (4, 6) and forming triangles of given area with the coordinate axes (each line forms a triangle of fixed area with the axes).P,Q,R&S needs to be compared with the choice 1,2,3&4.

(P) Number of lines which form an area of 36 sq. units =    (1) 1
(Q) Number of lines which form an area of 48 sq. units =    (2) 2
(R) Number of lines which form an area of 60 sq. units =    (3) 3
(S) Number of lines which form an area of 40 sq. units =    (4) 4

My approach is as follow the lines that passes through (4,6) is y=mx+6-4m

The intercepts of the lines on co-ordinate axes is $(\frac{4m-6}{m},0)$ and$(0,6-4m)$

Area is $A=|\frac{1}{2}*\frac{(4m-6)^2}{m}|$ after this I am little confused regarding the number of lines

Samar Imam Zaidi
  • 8,800
  • $8(m-\frac32)^2=\pm Am$ is quadratic, $D=(A\pm 24)^2-4\cdot 8\cdot 18=A(A\pm48)$, $D>0$: $2$ solutions, $D=0$: $1$ solution, $D<0$: $0$ solutions. the $\pm$ sign is to be considered rather accurately only. – Alexey Burdin Jun 15 '20 at 12:03

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