By signing in you confirm you are 18 or older, or have your parent’s or guardian’s consent.

NDA 1 2025 Maths, question 57

Topic: Analytical geometry

Q57

A square is inscribed in a circle x2+y2+2x+2y+1=0x^{2} + y^{2} + 2x + 2y + 1 = 0 and its sides are parallel to coordinate axes. Which one of the following is a vertex of the square?

  1. (a) (−2,2)( - 2,2)
  2. (b) (−2,−2)( - 2, - 2)
  3. (c) (−1+1/2,−1−1/2)\left( - 1 + 1/\sqrt{2}, - 1 - 1/\sqrt{2} \right)Correct answer
  4. (d) None of the above

Solution.

Answer: (c) (−1+1/2,−1−1/2)\left( - 1 + 1/\sqrt{2}, - 1 - 1/\sqrt{2} \right)

  1. Complete squares: (x+1)2+(y+1)2=1(x + 1)^{2} + (y + 1)^{2} = 1.
  2. The circle has centre (−1,−1)( - 1, - 1) and radius 1.
  3. For an axis-parallel inscribed square, each vertex is displaced by (±h,±h)( \pm h, \pm h) from the centre, where 2h2=12h^{2} = 1.
  4. Thus h=1/2h = 1/\sqrt{2}, and the vertex in option (c) is one of the four possibilities.