NIMCET 2011 — Mathematics PYQ

1. Find the points where the lines intersect the axes

• Line 1: $2x+3y-6=0$

  • X-intercept (put $y=0$): $2x=6⟹x=3$. Point $A=(3,0)$

  • Y-intercept (put $x=0$): $3y=6⟹y=2$. Point $B=(0,2)$

• Line 2: $9x+6y-18=0$

  • X-intercept (put $y=0$): $9x=18⟹x=2$. Point $C=(2,0)$

  • Y-intercept (put $x=0$): $6y=18⟹y=3$. Point $D=(0,3)$

2. Understand the Concyclic Condition

The problem states these four points—$A(3,0)$, $B(0,2)$, $C(2,0)$, and $D(0,3)$—lie on the same circle. We need to find the center $(h,k)$ of this circle.

3. Use the Perpendicular Bisector Property

The center of a circle lies at the intersection of the perpendicular bisectors of any two chords.

• Chord AC (on the x-axis):

  The points are $(3,0)$ and $(2,0)$.

  The perpendicular bisector of a horizontal segment is a vertical line passing through the midpoint.

  $$\text{Midpoint of AC}=\left(\frac{3+2}{2},0\right)=(2.5,0)$$

  Equation of the perpendicular bisector: $x=\frac{5}{2}$

• Chord BD (on the y-axis):

  The points are $(0,2)$ and $(0,3)$.

  The perpendicular bisector of a vertical segment is a horizontal line passing through the midpoint.

  $$\text{Midpoint of BD}=\left(0,\frac{2+3}{2}\right)=(0,2.5)$$

  Equation of the perpendicular bisector: $y=\frac{5}{2}$

4. Final Answer

The center $(h,k)$ is the intersection of $x=5/2$ and $y=5/2$.

The correct option is (d).