Question

The line L passes through the point (9, 4) and has negative slope. The area of the triangle that L cuts from the first quadrant is 72. Find an equation of the line L

Answer 1

`\sf\\\textsf{From the figure,}\\\textsf{Area of triangle = }(1)/(2)* a* b=(ab)/(2)\\\textsf{or, 72 = }(ab)/(2)\\\\\textsf{or, 144=ab}`

`\sf\\\textsf{or, }b=(144)/(a).......(1)`

`\sf\\\textsf{Here from figure, y-intercept = b, x-intercept =a }\\\textsf{Using double-intercept form, the equation of the line L is:}\\\\(x)/(a)+(y)/(b)=1\\\\\textsf{or, }bx + ay = ab.........(2)\\\textsf{According to the question, Line L passes through (9, 4).}\\\therefore\ 9b+4a=144..........(3)`

`\sf\\\textsf{Substituting equation (1) in equation (3),}\\9((144)/(a))+4a=144\\\\\textsf{or, } (1296)/(a)+4a=144\\\\\textsf{or, }(1296+4a^2)/(a)=144\\\\\textsf{or, }4a^2-144a+1296=0\\\textsf{or, }a^2-36a+324=0\\\textsf{or, }a^2-2.a.(18)+18^2=0\\\textsf{or, }(a-18)^2=0\\\textsf{or, }a=18`

`\sf\\\textsf{Equation (1) becomes: }\\\\b=(144)/(18)\\\\\textsf{or, }b=8`

`\sf\\\textsf{Equation (2) becomes}\\8x+18y=144\\\textsf{or, }4x+9y=72\textsf{ is the required equation of line L.}`