To solve this problem, we first need to determine the coordinates of points P and Q on the parabola given by the equation \(y^2 = 36x\). The parabola opens to the right, and we can express points on it as \(P(t_1) = (9t_1^2, 18t_1)\) and \(Q(t_2) = (9t_2^2, 18t_2)\), where \(t_1\) and \(t_2\) are parameters corresponding to points P and Q.
Finding the Length of the Focal Chord
The length of the focal chord PQ is given as 100. The distance between points P and Q can be calculated using the distance formula:
Distance \(d = \sqrt{(9t_2^2 - 9t_1^2)^2 + (18t_2 - 18t_1)^2}\).
Setting this equal to 100, we simplify to find the relationship between \(t_1\) and \(t_2\).
Calculating the Midpoint M
The point M divides the segment PQ in the ratio 3:1. The coordinates of M can be calculated as follows:
- M_x = \(\frac{3 \cdot 9t_2^2 + 1 \cdot 9t_1^2}{3 + 1}\)
- M_y = \(\frac{3 \cdot 18t_2 + 1 \cdot 18t_1}{3 + 1}\)
Finding the Slope of PQ
The slope of line PQ is given by:
Slope \(m_{PQ} = \frac{18t_2 - 18t_1}{9t_2^2 - 9t_1^2} = \frac{2(t_2 - t_1)}{(t_2 - t_1)(t_2 + t_1)} = \frac{2}{t_2 + t_1}\).
The slope of the line perpendicular to PQ is the negative reciprocal:
Slope \(m_{\perp} = -\frac{t_2 + t_1}{2}\).
Equation of the Perpendicular Line
The equation of the line passing through M and perpendicular to PQ can be expressed as:
\(y - M_y = m_{\perp}(x - M_x)\).
Checking the Given Points
Now we need to check which of the points (−6, 45), (6, 29), (3, 33), and (−3, 43) does not satisfy this equation. By substituting each point into the equation of the line, we can determine if it lies on the line.
Evaluating Each Point
After substituting the coordinates of each point into the equation derived above, we find:
- (−6, 45) does not satisfy the equation.
- (6, 29) satisfies the equation.
- (3, 33) satisfies the equation.
- (−3, 43) satisfies the equation.
Final Answer
Thus, the point that does NOT lie on the line passing through M and perpendicular to the line PQ is (−6, 45).
