Question

Please tell the answer with proper explanation. I am confused...​

Answer 1

Question:

The point which is equidistant from the points ( 0, 0 ), ( 0, 8 ) & ( 4, 6 ) is

1) $\displaystyle{\sf\:\left(\:\dfrac{1}{2}\;,\;-\:4\:\right)}$

2) $\displaystyle{\sf\:\left(\:-\:\dfrac{1}{2}\;,\;4\:\right)}$

3) $\displaystyle{\sf\:\left(\:\dfrac{1}{2}\;,\;4\:\right)}$

4) $\displaystyle{\sf\:\left(\:-\:\dfrac{1}{2}\;,\;-\:4\:\right)}$

Answer:

The point which is equidistant from the given points is

$\displaystyle{\boxed{\red{\sf\:\left(\:\dfrac{1}{2}\:,\:4\:\right)}}}$

Option 3)

Step-by-step-explanation:

Let the given points be P, Q & R.

The coordinates of the given points are

P ≡ ( 0, 0 ) ≡ ( x₁, y₁ )

Q ≡ ( 0, 8 ) ≡ ( x₂, y₂ )

R ≡ ( 4, 6 ) ≡ ( x₃, y₃ )

Let the point A be equidistant from the given point.

A ≡ ( x, y )

∴ PA = QA = RA

∴ PA = QA

Now, by distance formula,

√[ ( x - x₁ )² + ( y - y₁ )² ] = √[ ( x - x₂ )² + ( y - y₂ )² ]

⇒ √[ ( x - 0 )² + ( y - 0 )² ] = √[ ( x - 0 )² + ( y - 8 )² ]

⇒ ( x - 0 )² + ( y - 0 )² = ( x - 0 )² + ( y - 8 )² $\qquad\:\dots$ [ Squaring each term ]

⇒ x² + y² = x² + y² - 16y + 8²

⇒ x² + y² - x² - y² + 16y = 64

⇒ 0 + 16y = 64

⇒ 16y = 64

$\displaystyle{\implies\sf\:y\:=\:\cancel{\dfrac{64}{16}}}$

$\displaystyle{\implies{\boxed{\pink{\sf\:y\:=\:4\:}}}}$

Now,

PA = RA

Now, by distance formula,

√[ ( x - x₁ )² + ( y - y₁ )² ] = √[ ( x - x₃ )² + ( y - y₃ )² ]

⇒ √[ ( x - 0 )² + ( y - 0 )² ] = √[ ( x - 4 )² + ( y - 6 )² ]

⇒ ( x - 0 )² + ( y - 0 )² = ( x - 4 )² + ( y - 6 )² $\qquad\:\dots$ [ Squaring each term ]

⇒ x² + y² = x² - 8x + 4² + y² - 12y + 6²

⇒ x² + y² = x² - 8x + 16 + y² - 12y + 36

⇒ x² + y² - x² - y² + 8x + 12y = 16 + 36

⇒ 8x + 12y = 52

⇒ 2x + 3y = 13 $\qquad\:\dots$ [ Dividing by 4 ]

⇒ 2x + 3 * 4 = 13

⇒ 2x + 12 = 13

⇒ 2x = 13 - 12

⇒ 2x = 1

$\displaystyle{\implies\boxed{\blue{\sf\:x\:=\:\dfrac{1}{2}}}}$

∴ The point which is equidistant from the given points is

$\displaystyle{\underline{\boxed{\red{\sf\:\left(\:\dfrac{1}{2}\:,\:4\:\right)}}}}$

Answer 2

Step-by-step explanation:

1/2,4 ❤ .jaiic