Question

If the slope of the line (2-3)x+(2+3)y+3=0 is a-b then a^2+b=

• A. 40
• B. 49
• C. 53
• D. 14

Hint:

When rationalizing fractions like 3-23+2 where the squares of the numbers differ by exactly 1 (3 - 2 = 1), the denominator disappears entirely upon rationalization, and the expression simplifies instantly to (numerator)^2. Recognizing these structural numbers saves considerable algebraic expansion steps.

Answer 1

The Correct Option is C

Concept: For any standard linear straight-line equation written in general form as Ax + By + C = 0, its mathematical slope (m) is given by the negative ratio of the x-coefficient to the y-coefficient: \[ m = -\frac{A}{B} \] To simplify expressions involving radicals in the denominator, we rationalize the denominator by multiplying both the numerator and the denominator by its conjugate.

Step 1: Identify the coefficients and calculate the basic expression for the slope. From the given equation (2-3)x+(2+3)y+3=0:

• A = 2-3

• B = 2+3
Therefore, the slope m is: \[ m = -\frac{\sqrt{2}-\sqrt{3}}{\sqrt{2}+\sqrt{3}} = \frac{-(\sqrt{2}-\sqrt{3})}{\sqrt{2}+\sqrt{3}} = \frac{\sqrt{3}-\sqrt{2}}{\sqrt{2}+\sqrt{3}} \]

Step 2: Rationalize the denominator of the slope expression. The conjugate of the denominator (3+2) is (3-2). Multiplying the numerator and denominator by this conjugate: \[ m = \frac{(\sqrt{3}-\sqrt{2})(\sqrt{3}-\sqrt{2})}{(\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})} \] Apply the difference of squares identity (x+y)(x-y) = x^2 - y^2 to the denominator: \[ \text{Denominator} = (\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1 \] Now expand the numerator using the identity (x-y)^2 = x^2 - 2xy + y^2: \[ \text{Numerator} = (\sqrt{3}-\sqrt{2})^2 = (\sqrt{3})^2 - 2(\sqrt{3})(\sqrt{2}) + (\sqrt{2})^2 \] \[ \text{Numerator} = 3 - 2\sqrt{6} + 2 = 5 - 2\sqrt{6} \] Putting it back together: \[ m = \frac{5 - 2\sqrt{6}}{1} = 5 - 2\sqrt{6} \]

Step 3: Adjust the form to match the given comparison expression a - b. We need to rewrite 26 entirely under a single radical sign: \[ 2\sqrt{6} = \sqrt{2^2 \times 6} = \sqrt{4 \times 6} = \sqrt{24} \] Thus, the calculated slope is: \[ m = 5 - \sqrt{24} \]

Step 4: Compare coefficients to find the values of a and b, then evaluate the final expression. By comparing 5 - 24 directly with a - b: \[ a = 5 \quad \text{and} \quad b = 24 \] Now, compute the target value a^2 + b: \[ a^2 + b = (5)^2 + 24 = 25 + 24 = 49 \] *Note: Let's double check our calculations. Ah, 25 + 24 = 49. The value is 49.* Let us check the option values provided: Option (B) is 49. Let's re-verify matching steps: Comparing a - b with 5 - 24 yields a = 5, b = 24. a^2 + b = 25 + 24 = 49. This perfectly matches Option (B).