Let a variable line passing through the centre of the circle x2 + y2 – 16x – 4y = 0, meet the positive co-ordinate axes at the point A and B. Then the minimum value of OA + OB, where O is the origin, is equal to
- 12
- 18
- 20
- 24
The Correct Option is B
Approach Solution - 1
Given Equation:
The equation of the line is given as: \[ (y - 2) = m(x - 8) \]
Step 1: Finding the x-intercept:
The x-intercept is found by setting \( y = 0 \), which gives: \[ \left( \frac{-2}{m} + 8 \right) \]
Step 2: Finding the y-intercept:
The y-intercept is found by setting \( x = 0 \), which gives: \[ (-8m + 2) \]
Step 3: Calculating \( OA + OB \):
The sum of distances \( OA \) and \( OB \) is given by: \[ OA + OB = \frac{-2}{m} + 8 - 8m + 2 \]
Step 4: Finding the minimum of the function:
We differentiate the function and set the derivative equal to zero: \[ f'(m) = \frac{2}{m^2} - 8 = 0 \]
Solving for \( m \): \[ m^2 = \frac{1}{4} \] Therefore: \[ m = -\frac{1}{2} \]
Step 5: Final Calculation of Minimum:
Finally, substituting \( m = -\frac{1}{2} \) into the function: \[ f\left( -\frac{1}{2} \right) = 18 \] Therefore, the minimum value is: \[ \text{Minimum} = 18 \]
Approach Solution -2
The equation of the circle is:
\[ x^2 + y^2 - 16x - 4y = 0 \]
Rewrite it in standard form by completing the square:
\[ (x - 8)^2 + (y - 2)^2 = 68 \]
The center of the circle is \( (8, 2) \).
Let the equation of the line passing through \( (8, 2) \) be:
\[ (y - 2) = m(x - 8) \]
Find the intercepts. For the x-intercept, set \( y = 0 \):
\[ 0 - 2 = m(x - 8) \] \[ x = \frac{-2}{m} + 8 \]
For the y-intercept, set \( x = 0 \):
\[ y - 2 = m(0 - 8) \] \[ y = -8m + 2 \]
Calculate \( OA + OB \). The distance \( OA + OB \) is given by the sum of the intercepts:
\[ OA + OB = \left| \frac{-2}{m} + 8 \right| + \left| -8m + 2 \right| \]
Define \( f(m) = \frac{-2}{m} + 8 - 8m + 2 \). To find the minimum value, take the derivative \( f'(m) \) and set it to zero:
\[ f'(m) = \frac{2}{m^2} - 8 = 0 \]
\[ \frac{2}{m^2} = 8 \] \[ m^2 = \frac{1}{4} \] \[ m = \pm \frac{1}{2} \]
Substitute \( m = -\frac{1}{2} \):
\[ f\left( -\frac{1}{2} \right) = 18 \]
Thus, the minimum value of \( OA + OB \) is:
18
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