Any point on the parabola whose focus is $(0, 1)$ and the directrix is $x + 2 = 0$ is given by

(t^{2} + 1, 2 t - 1)

No worries! We‘ve got your back. Try BYJU‘S free classes today!

(t^{2} + 1, 2 t + 1)

No worries! We‘ve got your back. Try BYJU‘S free classes today!

(t^{2}, 2 t)

No worries! We‘ve got your back. Try BYJU‘S free classes today!

(t^{2} - 1, 2 t + 1)

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

Open in App

Solution

The correct option is B $(t^{2} - 1, 2 t + 1)$
Given,

$f (0, 1), d (x + 2 = 0)$

Distance of any point on parabola and focus is equal to distance of point and directrix.

$f P = \sqrt{(h - 0)^{2} + (k - 1)^{2}} = \sqrt{h^{2} + k^{2} + 1 - 2 k}$

Distance of point $(h, k)$ and line $x + 2 = 0$

Using point line distance formula.

$d P = h + 2$

$\sqrt{h^{2} + k^{2} + 1 - 2 k} = h + 2$

$h^{2} + k^{2} + 1 - 2 k = h^{2} + 4 + 4 h$

$k^{2} - 2 k + 1 - 4 - 4 h = 0$

replacing $h \to x, k \to y$

$y^{2} - 2 y + 1 - 4 - 4 x = 0$

$(y - 1)^{2} = 4 (x + 1)$ $\dots (1)$

Let $Y = y - 1, X = x + 1$ then $(1)$ becomes

$Y^{2} = 4 a X^{2}$ Here $a = 1$

any point on this parabola will be of the form $(a t^{2}, 2 a t) = (t^{2}, 2 a t)$

$\Rightarrow X = t^{2} \Rightarrow x + 1 = t^{2} \Rightarrow x = t^{2} - 1$

$\Rightarrow Y^{2} = 2 t \Rightarrow y - 1 = 2 t \Rightarrow y = 2 t + 1$

$∴$ Any point on the parabola $(y - 1)^{2} = 4 (x + 1)$ is $(t^{2} - 1, 2 t + 1)$

Suggest Corrections

Similar questions

Q. Parametric point on the parabola whose focus is (0, 1) and the directrix is x + 2 =0 is

Any point on the parabola whose focus is (0, 1) and the directrix is x+2=0 is given by

Q. If

x = \sqrt{\frac{1 - t^{2}}{1 + t^{2}}}

and

y = \frac{\sqrt{1 + t^{2}} - \sqrt{1 - t^{2}}}{\sqrt{1 + t^{2}} + \sqrt{1 - t^{2}}},

then the value of

\frac{d^{2} y}{d x^{2}}

t = 0

is given by

Show that the point (x, y) given by $x = \frac{2 a t}{1 + t^{2}}$ and $y = \frac{a (1 - t^{2})}{1 + t^{2}}$ lies on a circle for all real values of that such that $- 1 \leq t \leq 1$ , where a is any given real numbers.

Find the equations of the altitudes of the triangle whose vertices are A (7, - 1), B(- 2, 8) and C (1, 2).

Q. Coordinates of any point on the parabola, whose focus is

(- \frac{3}{2}, - 3)

and the directrix is

2 x + 5 = 0

is given by

Join BYJU'S Learning Program

Any point on the parabola whose focus is (0,1) and the directrix is x+2=0 is given by

Any point on the parabola whose focus is $(0, 1)$ and the directrix is $x + 2 = 0$ is given by