If P(a^2, 2a) is a point on a line segment joining the points A(2,0) and B(0,4), What is the ratio of the distances AP and BP
Answers
The ratio of distances AP & BP = 1 : 1.
Given Points
- A(2,0)
- B(0,4)
Slope of the line,
Equation of the line,
y - 0 = - 2 ( x - 2)
y = - 2x + 4
2x + y - 4 = 0
From the question,
P ( a², 2a) lies on the above line,
⇒2a² + 2a - 4 = 0
⇒ a² + a - 2 = 0
⇒ a² + 2a - a - 2 = 0
⇒ a ( a + 2) - ( a + 2) = 0
⇒ (a - 1) ( a + 2) = 0
⇒ a = 1 or a = - 2
So P can be, ( 1, 2) or ( 4, - 4)
If P is (1, 2), P divides AB in the ratio m : n then,
⇒1 = m(0)+ n(2) / m + n
⇒m + n = 2n
⇒ m = n
⇒ m : n = 1 : 1
If P is (4,-4), Let P divides AB in the ratio of m : n then,
⇒ 4 = m(0)+ n(2) / m + n
⇒ 4m + 4n = 2n
⇒ 4m = - 2n
⇒ 2m = - n
⇒ m : n = - 1 : 2
Given that P lies on the line segment but If P is (4,-4), It becomes an external point.
Therefore, P is ( 1, 2) and Ratio of AP : BP = 1 : 1
Step-by-step explanation:
The ratio of distances AP & BP = 1 : 1.
Given Points
A(2,0)
B(0,4)
Slope of the line,
m = \frac{4 - 0}{ 0 - 2} = \frac{4}{ - 2 } = - 2m=
0−2
4−0
=
−2
4
=−2
Equation of the line,
y - 0 = - 2 ( x - 2)
y = - 2x + 4
2x + y - 4 = 0
From the question,
P ( a², 2a) lies on the above line,
⇒2a² + 2a - 4 = 0
⇒ a² + a - 2 = 0
⇒ a² + 2a - a - 2 = 0
⇒ a ( a + 2) - ( a + 2) = 0
⇒ (a - 1) ( a + 2) = 0
⇒ a = 1 or a = - 2
So P can be, ( 1, 2) or ( 4, - 4)
If P is (1, 2), P divides AB in the ratio m : n then,
⇒1 = m(0)+ n(2) / m + n
⇒m + n = 2n
⇒ m = n
⇒ m : n = 1 : 1
If P is (4,-4), Let P divides AB in the ratio of m : n then,
⇒ 4 = m(0)+ n(2) / m + n
⇒ 4m + 4n = 2n
⇒ 4m = - 2n
⇒ 2m = - n
⇒ m : n = - 1 : 2
Given that P lies on the line segment but If P is (4,-4), It becomes an external point.
Therefore, P is ( 1, 2) and Ratio of AP : BP = 1 : 1