A line passes through (2,2) and cuts a triangle of area 9 " units"^2 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

2 Answers
Mar 6, 2018

Sum of slopes is -2.5 and answer is (A).

Explanation:

A line passing through point (2,2) and having a slope m has an equation

y-2=m(x-2)

or mx-y=2m-2=2(m-1)

or (mx)/(2(m-1))-y/(2(m-1))=1

or x/((2(m-1))/m)+y/(-2(m-1))=1 (intercept form of equation)

Observe that such a line forming a triangle in Q1, will have a negative slope.

and its x-intercept is (2(m-1))/m and y-intercept is -2(m-1)

and hence area of triangle so formed is

1/2xx(2(m-1))/mxx-2(m-1) and as area is 9 we have

-2(m-1)^2/m=9

or -2(m-1)^2=9m

or -2m^2+4m-2=9m

or 2m^2+5m+2=0

or (2m+1)(m+2)=0

i.e. m=-1/2 or m=-2

Hence sum of slopes is -2.5 i.e. answer is (A)

and lines are x+2y=6 or 2x+y=6

graph{(x+2y-6)(2x+y-6)=0 [-6.69, 13.31, -2.62, 7.38]}

Mar 6, 2018

A) -2.5.

Explanation:

Let, the X-intercept and Y-intercept of the line, say l, through the

point P(2,2) be a and b, resp.

Clearly, l : x/a+y/b=1.

P in l"...[Given] "rArr 2/a+2/b=1.........(star).

Observe that, l makes, with the Axes, such a right-triangle,

of which, the lengths of the sides making the right angle are

a and b.

Hence, the area of the right-triangle is 1/2ab.

Knowing that, 1/2ab=9, b=18/a.

Then, (star) rArr 2/a+2*a/18=1, or, a^2-9a+18=0.

rArr a=6, or 3, & :., b=3, or 6.

These give 2 values of slope m=-b/a of l, namely,

m_1=-1/2, or m_2=-2," giving the sum of slopes, "-2.5.