Question

A line passes through (2,2) and cuts a triangle of area 9 from the first quadrant. The sum of all possible values for the slope of such a line, is?

A) -2.5
B) -2
C) -1.5
D) -1

Answer 1

Answer is `(A)`.

Explanation:

Let the slope of the line be `m`. As it passes through `(2,2)`, its equattion is

`y-2=m(x-2)` or `y=mx-2m+2`

Now, area of triangle in such a case is half the product of `x`-intercept and `y`-intercept.

`x`-intercept is given by putting `y=0` i.e. `mx-2m+2=0` or `x=(2m-2)/m` and `y`-intercept is given by putting `x=0` i.e. `y=-2m+2`.

Hence, if area of triangle is `A`

`2A=(-2m+2)(2m-2)/m=(-4m^2+8m-4)/m`

or `18=(-4m^2+8m-4)/m`

or `18m=-4m^2+8m-4`

or `4m^2+10m+4=0`

As in a quadratic equation `ax^2+bx+c=0`, sum of roots is `-b/a`

sum of slopes is `-10/4=-2.5` and answer is `(A)`.

graph{(x+2y-6)(y+2x-6)=0 [-1, 13, -0.5, 6.5]}