Question

If (1, a), (b, 2) are conjugate points with respect to the circle \(x^2 + y^2 = 25\), then \(4a + 2b =\)

• A. \(25\)
• B. \(50\)
• C. \(100\)
• D. \(150\)

Hint:

For two points \((x_1, y_1)\) and \((x_2, y_2)\) to be conjugate with respect to a circle \(x^2 + y^2 + 2gx + 2fy + c = 0\), the key condition is \(x_1x_2 + y_1y_2 + g(x_1+x_2) + f(y_1+y_2) + c = 0\). This condition is a direct application of the polar equation, where the polar of one point passes through the other. For a circle centered at the origin \(x^2 + y^2 = r^2\), the condition simplifies to \(x_1x_2 + y_1y_2 = r^2\).

Answer 1

The Correct Option is B

Step 1: Understand the condition for conjugate points.
Two points \((x_1, y_1)\) and \((x_2, y_2)\) are said to be conjugate with respect to a circle \(x^2 + y^2 + 2gx + 2fy + c = 0\) if the polar of one point passes through the other point. The condition for conjugacy is \(x_1x_2 + y_1y_2 + g(x_1+x_2) + f(y_1+y_2) + c = 0\). 
Step 2: Identify the circle equation and the given points.
The given circle equation is \(x^2 + y^2 = 25\), which can be written as \(x^2 + y^2 - 25 = 0\). Comparing this to the general form \(x^2 + y^2 + 2gx + 2fy + c = 0\), we have:
\(g = 0\)
\(f = 0\)
\(c = -25\)
The two given points are \((x_1, y_1) = (1, a)\) and \((x_2, y_2) = (b, 2)\).
Step 3: Apply the conjugate points condition.
Substitute the coordinates of the points and the coefficients of the circle into the conjugacy condition:
\(x_1x_2 + y_1y_2 + g(x_1+x_2) + f(y_1+y_2) + c = 0\) \[ (1)(b) + (a)(2) + 0(1+b) + 0(a+2) + (-25) = 0 \] \[ b + 2a - 25 = 0 \] Rearrange the terms to find the relationship between \(a\) and \(b\): \[ 2a + b = 25 \] Step 4: Calculate the required expression \(4a + 2b\).
We need to find the value of \(4a + 2b\).
Notice that \(4a + 2b\) is simply \(2\) times the expression \(2a + b\): \[ 4a + 2b = 2(2a + b) \] Substitute the value of \(2a + b\) from Step 3: \[ 4a + 2b = 2(25) \] \[ 4a + 2b = 50. \]