Show that the chords of contacts of points on the line 2x – 3y + 4 = 0 with respect to the parabola y2 = 4ax pass through a fixed point.
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Show that the chords of contacts of points on the line 2x – 3y + 4 = 0 with respect to the parabola y2 = 4ax pass through a fixed point.
Show that the chords of contacts of points on the line 2x – 3y + 4 = 0 with respect to the parabola y2 = 4ax pass through a fixed point.
Let P(x1, y1) be a point on the line 2x – 3y + 4 = 0. Therefore
2x1 – 3y1 + 4 = 0 ….(1)
Now, the chord of contact of (x1, y1) with respect to y2 + 4ax is
yy1 – 2a(x + x1) = 0 ….(2)
From Eqs. (1) and (2), we get
yy1 – 2ax – a(3y1 – 4) = 0
y1(y – 3a) 2a(x – 2) = 0 …(3)
Eq. (3) represents the lines passing through the fixed point which is the intersection of the lines x = 2 and y = 3a. Hence, the fixed point is (2, 3a).