Math, asked by saumyasinha223, 10 months ago

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

Answered by HappiestWriter012
3

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 }  =  - 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

Answered by Rounakraj001
0

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

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