A straight line x/a -y/b =1 passes through the point (8,6) and cuts off a triangle of area 12 units from the axes of coordinates .find the equation of straight line

since (8,6) lies on it,

8/a - 6/b = 1
(8b - 6a)/(ab) = 1
8b - 6a = ab
8b - ab = 6a
b(8 - a) = 6a
b = 6a/(8-a) **

x/a - y/b = 1 is in the intercept form, that is
the x-intercept is (a,0)
the y-intercept is (0,b)
so the area of the triangle is (1/2)(ab)
(1/2)(ab) = 12
ab = 24 ***

sub ** into ***
a(6a)/(8-a) = 24
6a^2 = 192 - 24a
a^2 + 4a - 32 = 0
(a+8)(a-4) = 0
a = -8 or a = 4

if a = 4, b = 6
x/4 - y/6 = 1 ---> 6x - 4y = 24 or 3x - 2y = 12

if a = -8, b = -3
x/-8 - y/-3 = 1 ---> -3x + 8y = 24 or 3x - 8y = -24

Very useful

Ty

To find the equation of the straight line, we can follow these steps:

1. First, let's find the intercepts of the line with the x and y axes.
- To find the x-intercept, we can set y = 0 and solve for x:
x/a - 0/b = 1
x/a = 1
x = a

- To find the y-intercept, we can set x = 0 and solve for y:
0/a - y/b = 1
-y/b = 1
y = -b

So, the x-intercept is (a, 0) and the y-intercept is (0, -b).

2. Now, let's find the equation of the line using the two intercept points:
- We know that the slope of the line passing through (a, 0) and (0, -b) is given by:
m = (change in y) / (change in x)
m = (-b - 0) / (0 - a)
m = b/a

- Using the slope-intercept form of a line, the equation of the straight line passing through the intercepts can be written as:
y = mx + c, where m is the slope and c is the y-intercept.

3. To find the y-intercept, c, we can substitute the coordinates of the point (8, 6) into the equation of the line:
6 = (b/a)(8) + c
6 = 8b/a + c

4. We can also use the fact that the line cuts off a triangle of area 12 units from the axes to find another equation.
- The triangle's area can be calculated as half the product of the lengths of the two intercepts:
Area of triangle = (1/2) * (a * b) = 12

5. Now, we have two equations:
6 = 8b/a + c (Equation 1)
ab = 24 (Equation 2)

6. We have two equations with two variables (a, b) and two unknowns (c, b). We can solve them simultaneously to find the values of a, b, and c. However, since we do not have a specific value for a or b, we have an infinite number of possible solutions. Nevertheless, we can still express the equation of the line in terms of a, b, and c:
y = (b/a)x + (6 - (8b/a))

So, the equation of the straight line in terms of a, b, and c is:
y = (b/a)x + (6 - (8b/a))