Question

The locus of a point, which moves such that the sum of squares of its distances from the points (0, 0), (1, 0), (0, 1), (1, 1) is 18 units, is a circle of diameter $d$. Then $d^2$ is equal to _________.

Hint:

For $n$ points, the locus of a point such that the sum of the squared distances is constant is always a circle with its center at the centroid of the points.

Answer 1

Correct Answer: 16

Step 1: Understanding the Concept:
Let the moving point be $P(x, y)$. We set up the algebraic equation for the sum of the squares of distances and simplify it to the standard form of a circle $(x - h)^2 + (y - k)^2 = R^2$.
Step 2: Detailed Explanation:
Sum of squared distances:
$(x^2 + y^2) + ((x - 1)^2 + y^2) + (x^2 + (y - 1)^2) + ((x - 1)^2 + (y - 1)^2) = 18$.
Expanding:
$x^2 + y^2 + (x^2 - 2x + 1) + y^2 + x^2 + (y^2 - 2y + 1) + (x^2 - 2x + 1) + (y^2 - 2y + 1) = 18$.
$4x^2 + 4y^2 - 4x - 4y + 4 = 18$.
$4x^2 - 4x + 1 + 4y^2 - 4y + 1 + 2 = 18 \implies (2x - 1)^2 + (2y - 1)^2 = 16$.
Divide by 4:
$(x - 1/2)^2 + (y - 1/2)^2 = 4$.
This is a circle with radius $R = 2$.
Diameter $d = 2R = 4$.
$d^2 = 16$.
Step 3: Final Answer:
The value of $d^2$ is 16.