Find the equations of the straight lines each passing through the point (6,-2) and whose sum
of the intercept is 5.
Answers
EXPLANATION.
Equation of straight line passing through the points = (6,-2).
Sum of the intercept = 5.
As we know that,
Formula of :
Intercept form :
⇒ x/a + y/b = 1.
Sum of intercept = 5.
⇒ a + b = 5.
⇒ b = 5 - a.
⇒ 6/a + (-2)/(5 - a) = 1.
⇒ 6/a - 2/(5 - a) = 1.
⇒ 6(5 - a) - 2(a)/(a)(5 - a) = 1.
⇒ 30 - 6a - 2a/(5a - a²) = 1.
⇒ 30 - 8a = 5a - a².
⇒ a² - 5a - 8a + 30 = 0.
⇒ a² - 13a + 30 = 0.
Factorizes the equation into middle term splits, we get.
⇒ a² - 10a - 3a + 30 = 0.
⇒ a(a - 10) - 3(a - 10) = 0.
⇒ (a - 3)(a - 10) = 0.
⇒ a = 3 and a = 10.
When a = 3.
⇒ b = 5 - a.
⇒ b = 5 - 3 = 2.
When a = 10.
⇒ b = 5 - a.
⇒ b = 5 - 10 = - 5.
Put the value in the intercept form, we get.
⇒ x/a + y/b = 1.
When = (3,2).
⇒ x/3 + y/2 = 1.
⇒ 2x + 3y/6 = 1.
⇒ 2x + 3y = 6.
When = (10,-5).
⇒ x/10 + y/(-5) = 1.
⇒ x/10 - y/5 = 1.
⇒ x - 2y/10 = 1.
⇒ x - 2y = 10.
MORE INFORMATION.
Slope of a line.
(1) = m = tanθ, where θ is the angle made by a line with the positive direction of x-axes in anticlockwise.
(2) = The slope of a line joining two points (x₁, y₁) and (x₂, y₂) is given by : m = (y₂ - y₁)/(x₂ - x₁).
Given :-
Point (6,-2)
Sum of intercep = 5
To Find :-
Equation
Solution :-
a + b = 5
b = 5 - a(1)
Now
x/a + y/b = 1
Here
x = 6
a = a
y = -2
b = b
6/a + -2/b = 1
6/a + -2/5 - a = 1
30 - 6a - 2a/a × 5 - a = 1
30 - 8a/a × 5 - a = 1
30 - 8a = 5a - a²
30 - 8a - 5a - a² = 0
a² - 5a - 8a + 30 = 0
a² - 13a + 30 = 0
a² - (10a + 3a) + 30 = 0
a² - 10a - 3a + 30 = 0
a(a - 10) - 3(a + 10) = 0
(a - 10)(a - 3) = 0
Either
a - 10 = 0
a = 0 + 10
a = 10
or
a - 3 = 0
a = 0 + 3
a = 3
Now
a = 10
b = 5 - a
b = 5 - 10
b = -5
Equation = 2x + 3y = 6
a = 3
b = 5 - a
b = 5 - 2
b = 3
Equation = x - 2y = 10