Question

Limits and Geometry Let \(P \left( a , a ^ { 2 } \right)\) be a point on the parabola \(y = x ^ { 2 } , a > 0 .\) Let \(O\) be the origin and \(( 0 , b )\) the \(y\) -intercept of the perpendicular bisector of line segment \(O P .\) Find \(\lim _ { P \rightarrow O } b\)

Short answer

The limit of b as P approaches O is -1/2.

Step-by-step solution

Find the Slope of Line OP

The slope of a line given two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(m = (y_2 - y_1) / (x_2 - x_1)\). Here, \((x_1, y_1)=(a, a^2)\) and \((x_2, y_2)=(0, 0)\), so the slope of line OP is \(m_{OP} = (0 - a^2) / (0 - a) = -a\)

Calculate the Slope of the Perpendicular Bisector

The slope of a line perpendicular to another line with slope \(m\) is \(-1/m\). Therefore, the slope of the perpendicular bisector is \(m_{bisector} = -1 / m_{OP} = 1/a\)

Construct Equation of the Perpendicular Bisector

The point-slope form of a line is given by \(y - y_1 = m (x - x_1)\), where \((x_1, y_1)\) is a point on the line and \(m\) is the slope of the line. The midpoint of the line segment OP, which is also a point on the bisector, is \((a/2, a^2/2)\). Plugging these into the formula gives the equation of the bisector as \(y - a^2/2 = (1/a)(x - a/2)\)

Find y-intercept of the Perpendicular Bisector

For the y-intercept, set \(x = 0\) in the equation of the line. This yields \(b = y = a^2/2 - a/2\)

Compute the Limit of b as P Approaches to O

As \(P \rightarrow O\), this means that \(a \rightarrow 0\). Therefore, we need to calculate \(\lim _ { a \rightarrow 0 } b\). Replacing b with its equation from Step 4 gives \(\lim _ { a \rightarrow 0 } (a^2/2 - a/2)\), which equals -1/2