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 d² is equal to _____________.

Solution

Maths sir · 4 steps

Write the four squared distances, add them, and the x² + y² terms show you a circle.

1. Set up the sum. Let the moving point be P(x, y). Write the square of its distance from each of the four points and set the total equal to 18.

   (x²+y²) + ((x-1)²+y²) + (x²+(y-1)²) + ((x-1)²+(y-1)²) = 18

2. Expand and collect. There are four x² and four y² terms. The -2x and -2y terms each appear twice, and the constants add to 4. So 4x² + 4y² - 4x - 4y + 4 = 18, which divides by 4 to give x² + y² - x - y - 7/2 = 0.
3. Read the circle. The centre is (1/2, 1/2), the centre of the square. Then r² = 1/4 + 1/4 + 7/2 = 4, so r = 2.
4. So. The diameter is d = 2r = 4, so d² = 16.

The common slip: 4. Stopping at r² = 4, or at d = 4, gives 4. The question asks for d² = (2r)² = 16.

The idea to keep. For a locus, call the point (x, y) and turn the condition into an equation. If x² and y² come with equal coefficients and there is no xy term, divide by that coefficient and read off the centre and radius.

Answer from the NTA final answer key. Solution by PrepWiser.