Question

Let PQ be a focal chord of the parabola y² = 36x of length 100, making an acute angle with the positive x-axis. Let the ordinate of P be positive and M be the point on the line segment PQ such that PM:MQ = 3:1. Then which of the following points does \(\underline{NOT}\) lie on the line passing through M and perpendicular to the line PQ? 

• A. (3,33)
• B. (-6, 45)
• C. (-3, 43)
• D. (6,29)

Answer 1

The Correct Option is C

Parametric form and focal chord:
The parabola is \(y^2 = 36x\). Its focus is \((9,0)\). A chord through the focus is a focal chord. In the standard form \(y^2 = 4ax\) with \(a=9\), points are \(\bigl(9t^2,\,18t\bigr)\). A known result: if \(P\) corresponds to parameter \(t_1\) and \(Q\) to \(t_2,\) then \(PQ\) is a focal chord iff \(t_1\,t_2 = -1.\) 

Coordinates of \(P\) and \(Q\):
Let \[ P = \bigl(9t^2,\,18t\bigr), \quad Q = \Bigl(9\bigl(-\tfrac{1}{t}\bigr)^2,\,18\bigl(-\tfrac{1}{t}\bigr)\Bigr), \] so that \(t \cdot \bigl(-\tfrac{1}{t}\bigr) = -1\). Imposing \(PQ=100\) and the condition that \(P\) has positive ordinate, we arrive (from the given steps or direct calculation) at \[ P = (81,\,54), \quad Q = (1,\,-6). \] They do lie on \(y^2=36x\), pass through \((9,0)\), and are 100 units apart. 

Point \(M\) such that \(PM:MQ = 3:1\):
Since \(M\) divides \(PQ\) in the ratio \(3:1\), it is found using the section formula: \[ M = P + \frac{3}{4}\,(Q - P) = (81,\,54) + \frac{3}{4}\,\bigl((1,\,{-6}) - (81,\,54)\bigr) = (21,\,9). \] 

Line perpendicular to \(PQ\) through \(M\):
The slope of \(PQ\) is \[ \frac{54 - (-6)}{81 - 1} = \frac{60}{80} = \tfrac{3}{4}. \] Hence a line perpendicular to \(PQ\) has slope \(-\tfrac{4}{3}\). The equation of the perpendicular through \(M=(21,9)\) is: \[ y - 9 = -\tfrac{4}{3}\,\bigl(x - 21\bigr), \] so \[ y = -\tfrac{4}{3}\,x + 37. \] 

Checking given points: We substitute each point \((x,y)\) into \(y = -\tfrac{4}{3}x + 37\):

\((3,33)\): \(-\tfrac{4}{3}\cdot 3 + 37 = -4 + 37 = 33.\) Matches.

\((6,29)\): \(-\tfrac{4}{3}\cdot 6 + 37 = -8 + 37 = 29.\) Matches.

\((-6,45)\): \(-\tfrac{4}{3}\cdot (-6) + 37 = 8 + 37 = 45.\) Matches.

\(\boxed{(-3,43)}\): \(-\tfrac{4}{3}\cdot (-3) + 37 = 4 + 37 = 41,\) not \(43.\)

Hence \((-3,43)\) does not lie on the required perpendicular, so it is the correct choice.