Find the equation of the line through the point (3 4) which cuts from the
Answers
The general point slope form of a linear equation is y-y1=m(x-x1)
where m is the slope and (x1,y1)is the point the line goes through.
wkt
x1,y1=3,4
therefore
y-4=m(x-3)
y=m(x-3)+4
In order for this line to form a triangle in quadrant 1, the slope m must be negative. So m < 0. If m > 0 then we end up with an unbounded figure of infinite area.
If m < 0, then y=m(x-3)+4 forms a triangle with base b and height h. The area of the triangle is A = (b*h)/2. We need to find the base and height.
To find the height h, plug in x = 0 to find the y intercept
y=m(0-3)+4
y=-3m+4
So the y intercept is the point (0.-3m+4) where m is is some negative number.
So the height is h=-3m+4
To find the base b, we plug in y = 0 and solve for x. This yields the x intercept. The horizontal distance from the origin to the x intercept is equal to the base b
0=m(x-3)+4
mx=3m-4
x=(-4+3m)/m
therefore b=(-4+3m)/m
ARea of traingle
a=bh/2
a=[(-4+3m)/m*-3m+4]/2
on solving we get
a=(-9m^2+24m-16)/2m
f(x)=-9x^2+24x-16/2x
m<0
Because we replaced m with x, this means that x<0. Only focus on the portion that is to the left of the vertical y axis.
Use your calculator's "minimum" feature to find that the min point on the interval (-infinity, 0) is the point (-4/3, 24).
Note: -4/3 = -1.33 approximately.
The point P = (-1.33, 24) is marked on the graph above as that minimum point
m=-4/3
is the slope which produces the smallest area 24
substituting this in equation
we get
y=-4/3*x+8