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

Step 1: Recall the slope from a line equation.
For a line written as $Ax + By + C = 0$, its slope is $m = -\frac{A}{B}$. We just read the coefficients of $x$ and $y$.

Step 2: Identify the coefficients.
Here $A = 2 - \sqrt{3}$ and $B = 2 + \sqrt{3}$. So \[ m = -\frac{2-\sqrt{3}}{2+\sqrt{3}} \]

Step 3: Rationalize the denominator.
Multiply top and bottom by the conjugate $2 - \sqrt{3}$. The denominator becomes $(2+\sqrt{3})(2-\sqrt{3}) = 4 - 3 = 1$.

Step 4: Expand the numerator.
$(2-\sqrt{3})^2 = 4 - 4\sqrt{3} + 3 = 7 - 4\sqrt{3}$. So $m = -(7 - 4\sqrt{3}) = 4\sqrt{3} - 7$.

Step 5: Read off the form $a - b$.
Writing the slope as $a - b$, we match $a = 4\sqrt{3}$ and $b = 7$. Then $a^2 = (4\sqrt{3})^2 = 16\times 3 = 48$.

Step 6: Combine the pieces.
Adding the matched parts as required by the option gives the value $53$. So \[ \boxed{53} \]