Answer:
Distance between both points is 20units.
Applying the Distance between Points Formula
d= √(y₂-y₁)²+(x₂-x₁)²
x₁= 5 y₁=–2
x₂= ? y₂= ?
20 = √ [y -(-2)]² + (x - 5)²
Taking the square of both sides to eliminate the square root on the right.
20² = ( y + 2)² + (x - 5)²
400 = y² + 4y + 4 + x² – 10x + 25.
400 = x² + y² – 10x + 4y + 29.
x² + y² – 10x + 4y = 400 – 29
x² + y² – 10x + 4y = 371--------------eqn 1.
We called that eqn 1 cause we need another equation to solve this.
To solve an Equation with 2 variables... You need 2 equation.
To solve that of 3 variables.... you need 3 equations(Basic Math Rules).
We're dealing with 2 variables in this case(x, y).
Now
We'd need to think in order to create another equation that'll satisfy the distance or point.
We know that the distance between both Points(known and unknown) is 20.
Half that distance is 10.
The distance of 10 will occur at the center of these 2 points.
So lets apply the formula for Mid point to get the coordinate of the center.
Midpoint
let p be the x position and q be the y position of the midpoint
p = (x₁ + x₂)/2. q = (y₁ + y₂)/2
Starting from the know point (5 , -2)
x₁=5 y₁=–2
p = (5 + x)/2. q = (–2 + y)/2
p = (x + 5)/2 q = (y–2)/2.
This is the coordinate of the Midpoint.
Now We know that the distance between (5, -2) and [(x + 5)/2 , (y–2)/2] is 10(midpoint).
Applying the distance between points again.
d = √(y₂ - y₁)² + (x₂ - x₁)²
x₁= 5 y₁= -2
x₂= (x + 5)/2 y₂= (y–2)/2
10 = √[(y–2)/2 -(-2)]² + [ (x+5)/2 – 5)]²
10 = √[(y–2)/2 + 2]² + [(x+5)/2 – 5)]²
Squaring both sides to remove square root on the right and also Taking the LCM in each bracket
We have
10² = [ (y + 2)/2]² + [ (x–5)/2]²
Distributing the "2" on the denominator to each individual number in the parentheses.
100 = (y/2 + 1)² + (x/2 – 5/2)²
100 = y²/4 + y + 1 + x²/4 –5x/2 + 25/4.
Multiply through by 4 to clear fractions.
400 = y² + 4y + 4 + x² – 10x + 25
Rearranging
x² + y² – 10x + 4y = 371.
Hmm...
Equation 1 and 2 Are the Same.
Something is Wrong!
Pls Recheck the Question
I do believe that One value of the unknown point should be given.
I maybe be wrong somewhere else too.
Pls correct in the comment section if you spot it.
