SOLUTION: Find the equation of the line passing through the point (3,4) which cuts from the first quadrant of a triangle of minimum area.

Question 1082934: Find the equation of the line passing through the point (3,4) which cuts from the first quadrant of a triangle of minimum area.

Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
The general point slope form of a linear equation is

where

is the slope and

is the point the line goes through.

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We don't know the slope m, but we do know that

So,

and

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Plugging these x and y values into the point slope formula yields

Let's solve for y

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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 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

We stop here because we don't know m yet. If we did, we can replace m with the value and simplify.

So the y intercept is the point

where m is is some negative number.

So the height is

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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.

The x intercept is

where m is is some negative number.

The base is

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From the last two sections above, we found the following

base =

height =

The area of the triangle is therefore

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Replace the A with f(x). Replace the m with x. We end up with this function.

Graphing said function produces

Image generated by GeoGebra (free graphing software).

Now recall that . Because we replaced m with x, this means that . 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.

So is the slope which produces the smallest area 24.

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Plug in and simplify

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The final answer is
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