Find the equation of a line which cuts off intercepts on the axes whose sum and product are 1 and-6, respectively?
Thanks it was an easy method..
To find the equation of a line, we need to determine its slope and y-intercept.
Let's assume that the line intersects the x-axis at point (a, 0) and the y-axis at point (0, b). The product of the intercepts is given by ab = -6, and the sum of the intercepts is a + b = 1.
To find the slope, we can use the formula: slope (m) = (change in y) / (change in x).
The change in y is given by b - 0 = b, and the change in x is given by 0 - a = -a. So the slope is m = b / -a.
We can also find the slope using the y-intercept (0, b) and any other point on the line (a, 0). The slope, in this case, is m = (0 - b) / (a - 0) = -b / a.
Equating the two expressions for slope, we have b / -a = -b / a.
Multiplying both sides by -a, we get b^2 = -1.
Since the product of the intercepts (ab = -6) is negative, we have one positive intercept (a or b) and one negative intercept. Thus, b = 1 and a = -6.
Now that we have the slope and y-intercept, we can write the equation in slope-intercept form y = mx + b.
Plugging in the values, we get y = (-1/6)x + 1 as the equation of the line.
To find the equation of a line that cuts off intercepts on the axes, we need to determine the coordinates of these intercepts, and then use them to write the equation of the line.
Let's assume that the intercepts on the x-axis and y-axis are (a, 0) and (0, b) respectively.
Given that the sum of the intercepts is 1 and the product is -6, we can write the following equations:
a + b = 1 (equation 1)
ab = -6 (equation 2)
To solve this system of equations, we can use substitution or elimination method.
Let's solve it using the substitution method:
From equation 1, we can express 'a' in terms of 'b' as a = 1 - b.
Substituting this value in equation 2, we get:
(1 - b) * b = -6
Expanding and rearranging the equation, we have:
b - b^2 = -6
b^2 - b + 6 = 0
Now, we can solve this quadratic equation to find the value of 'b'.
Using the quadratic formula, we have:
b = (-(-1) ± sqrt((-1)^2 - 4*1*6)) / (2*1)
b = (1 ± sqrt(1 - 24)) / 2
b = (1 ± sqrt(-23)) / 2
Since we are dealing with a line cutting off intercepts on the axes, the b values cannot be complex numbers. Therefore, the discriminant (-23) is negative, meaning there are no real solutions.
This indicates that there is no line that satisfies the given conditions of intercepts whose sum and product are 1 and -6, respectively.
Recall the intercept form of a line: x/a + y/b = 1
Now, you have
a+b = 1
ab = -6
This can be solved, but you know the factors of 6 are 2 and 3, so you must have
a = -2
b = 3
So, the equation of the line is
x/-2 + y/3 = 1
Bow rearrange that into whichever form you like.