Question

If the distance between the point (1,2)&(3,8) is double of the distance of point (k+4,k) from the origin ,then find the value of k

Answer 1

Step-by-step explanation:

which is required ans of this question

Answer 2

Therefore the value of the k can be -1 or -3.

Given:

The points ( 1,2 ); ( 3,8 ); ( k+4, k ); ( 0,0 )

The distance between the point (1,2) and (3,8) is double the distance of the point (k+4,k) from the origin

To Find:

The value of 'k'.

Solution:

The given question can be solved as shown below.

The distance between ( x₁, y₁ ) and ( x₂, y₂ ) = √( x₂ - x₁ )² + ( y₂ - y₁ )²

The distance between ( 1,2 ) and ( 3,8 ) = √( 3 - 1 )² + ( 8 - 2 )²

⇒ The distance between ( 1,2 ) and ( 3,8 ) = √2² + 6² = √40

Now the distance between ( k+4, k ) and ( 0,0 ) = √ ( k+4 - 0 )² + ( k - 0)²

⇒ The distance between ( k+4, k ) and ( 0,0 ) = √k² + 8k + 16 + k² = √2k² + 8k + 16

Given that the  distance between the point (1,2)&(3,8) is double the distance of the point (k+4,k) from the origin

⇒ √40 = 2( √2k² + 8k + 16 )

Squaring on both sides,

⇒ 40 = 4( 2k² + 8k + 16 )

⇒ 10 = 2 ( k² + 4k + 8 )

⇒ 5 = k² + 4k + 8

⇒ k² + 4k + 3 = 0

⇒ k² + k + 3k + 3 = 0

⇒ k ( k + 1 ) + 3( k + 1 ) = 0

⇒ (k + 1 ) ( k + 3 ) = 0

⇒ k = -1 or -3

Therefore the value of the k can be -1 or -3.

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