MAH-CET 2026 — Mathematics PYQ

Step 1: Understand the geometric condition

We are looking for the family of circles where one diameter lies entirely along the line:

$$y=x$$

Since every diameter of a circle passes through its center, the center of the circle must lie on this line.

Step 2: Define the coordinates of the center

Let the center of the circle be $(h,k)$. Since the center lies on the line $y=x$, we can set:

$$k=h$$

Thus, the coordinates of the center can be written using a single parameter $h$ as:

$$\text{Center}=(h,h)$$

Step 3: Write the equation of the circle

Let the radius of the circle be $r$.

The standard equation of a circle with center $(h,h)$ and radius $r$ is:

$$(x-h{)}^2+(y-h{)}^2=r^2$$

Step 4: Identify the number of independent arbitrary constants

Let's count the arbitrary constants in the equation:

1. $h$ (determines the position of the center along the line $y=x$)

2. $r$ (determines the size/radius of the circle)

Since $h$ and $r$ can vary independently to give different circles of this family, there are exactly $2$ independent arbitrary constants.

Conclusion

Since the number of independent arbitrary constants is $2$, the order of the resulting differential equation must be $2$.

Correct Answer:

(b) 2