A point P moves on the parabola x2 – 2x + 4y – 11 = 0. The equatio

1.	A point P moves on the parabola x2 – 2x + 4y – 11 = 0. The equation of the locus of mid-point of line segment PS is (where S is the focus of the parabola)
Answer» x² – 2x + 4y – 11 = 0 ⇒ x² – 2x – 11 = -4y ⇒ x² – 2x + 1 - 12 = -4y ⇒ (x - 1)² = 4(-1)(y - 3) From here we can identify the focus. As in (x - h)²= 4p (y - k) the focus is given by (h, k + p) The directrix is y = k - p Here focus is (1, 4) and directrix is y = 4 Now it's easy to solve as the locus formed by that point will also be a parabola with same focus and distance will be half.

A point P moves on the parabola x2 – 2x + 4y – 11 = 0. The equation of the locus of mid-point of line segment PS is (where S is the focus of the parabola)

Answer»

x² – 2x + 4y – 11 = 0
⇒ x² – 2x – 11 = -4y
⇒ x² – 2x + 1 - 12 = -4y
⇒ (x - 1)² = 4(-1)(y - 3)
From here we can identify the focus.
As in (x - h)²= 4p (y - k) the focus is given by (h, k + p)
The directrix is y = k - p
Here focus is (1, 4) and directrix is y = 4

Now it's easy to solve as the locus formed by that point will also be a parabola with same focus and distance will be half.

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A point P moves on the parabola x2 – 2x + 4y – 11 = 0. The equation of the locus of mid-point of line segment PS is (where S is the focus of the parabola)

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