Operations On Real Numbers

The correct answer is 12
is interior of circle and is exterior of ellipse

∴ Area = πa² - πab = 30π .... (1)
Similarly and gives
πab - πb² = 30π .... (2)
Comparing (1) and (2) , and



and



= 12

Step 1: Start with the given equation

We are given two equations that describe the matrix :

Where is a matrix, and is the identity matrix.

Step 2: Express in terms of

We need to express in terms of . To begin, we square equation (1) to get the expression for :

Now, multiply both sides of this equation by to find the expression for :

Now, substitute into this equation to simplify:

Now, expand to get the expression for :

Now, substitute the expression for from equation (1) into this new expression for :

Now, expand this equation:

Step 3: Simplify the expression

From the equation for , we now have:

To make this consistent with equation (2), compare the coefficients of and :

For the coefficient of , we have:

For the coefficient of , we have:

Step 4: Conclusion

Thus, the values of and are:

Therefore, the correct values for and are and , as derived from the matrix equations.

Step 1: Using the given quadratic equations.

The roots of the first equation are and . Using the standard form of the sum and product of roots, we have:

, .

The roots of the second equation are and . Again, using the sum and product of roots, we get:

, .

Step 2: Solving for .

From equations (1) and (2), we know the following relationships for the sums and products of the roots:

To eliminate , subtract equation (1) from equation (2):

Simplify the right-hand side:

Thus,

Step 3: Using the product of roots.

Now, we use the product of roots from equations (1) and (2):

We know that and are the roots of the desired equation. Let’s compute the sum and product of these roots:

The sum of the roots is:

The product of the roots is:

Using the known relations for and , the required equation is:

Step 4: Conclusion.

Thus, the required equation is , and the correct answer is option (4).