JEE MATHEMATICS MCQ: CIRCLES

Solution approach

1. Correct Answer + Core Reason: 50. The value is found by using the condition for conjugate points with respect to a circle.
   
   Step-by-Step Solution:
   
   Step 1: The equation of the circle is ${x^2} + {y^2} = 25$.
   
   Step 2: The condition for two points $(x_1, y_1)$ and $(x_2, y_2)$ to be conjugate with respect to the circle ${x^2} + {y^2} = r^2$ is $x_1x_2 + y_1y_2 = r^2$.
   
   Step 3: Substitute the given points (1, a) and (b, 2) into the condition: $(1)(b) + (a)(2) = 25$.
   
   Step 4: Simplify the equation: $b + 2a = 25$.
   
   Step 5: Multiply the equation by 2: $2b + 4a = 50$.
   
   Step 6: Therefore, $4a + 2b = 50$.
   
   Why other options are wrong:
   - Option A: 25: This is incorrect because it only represents the value of b + 2a, not 4a + 2b.
   - Option C: 100: This is incorrect as it's double the correct answer, and doesn't follow from the conjugate point condition.
   - Option D: 150: This is incorrect as it's three times the value of b + 2a, and doesn't follow from the conjugate point condition.
   
   Key Concepts:
   - Circle equation
   - Conjugate points with respect to a circle
   
   Memory Aids:
   - Remember the condition for conjugate points: $x_1x_2 + y_1y_2 = r^2$.
   
   Quick Tips:
   - Apply the conjugate point condition directly to the given points and circle equation.