Q.12) Parabola

Q : The point (2, 1) divides a chord of the parabola y² = 4x in two
       equal line segments. Find the equation of the line, 
       containing that chord.

SOLUTION :

The equation of the required chord (bisected at a given point ) is given by S' = T

=> 1² - 4 * 2 = y * 1 - 2 (x + 2)
=> 1 - 8 = y - 2x - 4
=> -7 = y - 2x - 4
=> 2x - y - 3 = 0  => (Ans)

[ Here, S' = y² - 4x at (2, 1)
and T = tangent at (2, 1) ; ----> yy1 = 2a (x + x1) ]

Alternatively :

Let the co-ordinates of the extremities of the chord be P (t1²,2t1) and Q (t2²,2t2)

slope of chord PQ = 2(t2 - t1)/(t2² - t1²) ==> 2/(t1 + t2)
but (2t1 + 2t2)/2 = 1 {mid point formula}
=> 2(t1 + t2) = 2
=> (t1 + t2) = 1
slope = 2/1 = 2

equation of the required chord {with slpoe 2 and passes through (2, 1) } = y - 1 = (2)(x - 2)
=> 2x - y = 3 ------> (Ans)

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