Question

If the line \[ 2x+y=7 \] touches the curve \[ y=f(x) \] at \[ x=\frac12, \] then the sum of the lengths of the normal and subnormal is

• A. \(6+2\sqrt5\)
• B. \(6(2+\sqrt5)\)
• C. \(12+\sqrt5\)
• D. \(7+5\sqrt5\)

Hint:

If the tangent at a point has slope \(m\) and the point is \((x,y)\), then \[ \boxed{\text{Length of normal}=y\sqrt{1+m^2}} \] and \[ \boxed{\text{Length of subnormal}=|my|.} \]

Answer 1

The Correct Option is B

Step 1: Find the point of contact and slope. The tangent is \[ 2x+y=7 \] or \[ y=-2x+7. \] Hence, its slope is \[ m=-2. \] At \[ x=\frac12, \] the point of contact is \[ \left(\frac12,\, 7-2\cdot\frac12\right) = \left(\frac12,6\right). \]

Step 2: Find the lengths of the normal and subnormal. The length of the normal is \[ y\sqrt{1+m^2} = 6\sqrt{1+(-2)^2} = 6\sqrt5. \] The length of the subnormal is \[ |my| = |-2|\times6 = 12. \]

Step 3: Find the required sum. Therefore, \[ 6\sqrt5+12 = 6(2+\sqrt5). \] Hence, \[ \boxed{6(2+\sqrt5)}. \] Thus, \[ \boxed{(B)} \] is the correct answer.