Question

The centre of the circle passing through the point (0, 1) and touching the parabola y = x² at the point (2, 4) is ______.

विकल्प

• `[(-53)/10, 16/5]`

• `[6/5, 53/10]`

• `[3/10, 16/5]`

• `[(-16)/5, 53/10]`

Answer 1

The centre of the circle passing through the point (0, 1) and touching the parabola y = x² at the point (2, 4) is `underlinebb([(-16)/5, 53/10]`.

Explanation:

Circle passes through A(0, 1) and B(2, 4). So its centre is the point of intersection of the perpendicular bisector of AB and normal to the parabola at (2, 4).

Perpendicular bisector of AB;

`y - 5/2 = -2/3(x - 1)`

⇒ 4x + 6y = 19  ...(i)

Equation of normal to the parabola at (2, 4) is,

y – 4 = `-1/4(x - 2)`

⇒ x + 4y = 18  ...(ii)

∴ From (i) and (ii), x = `-16/5`, y = `53/10`

∴ Centre of the circle is `[-16/5, 53/10]`