P is a point on the parabola $y^{2} = 4 a x (a > 0)$ whose vertex is A. PA is produced to meet the directrix in D and M is the foot of the perpendicular from P on the directrix. If a circle is described on MD as a dimeter then it intersects the x-axis at a point whose co-ordinates are:

(-4a, 0 )

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

(-a, 0)

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

(-2a, 0)

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

(a , 0)

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

Open in App

Solution

The correct option is D (a , 0)
Let $P (a t_{1}^{2}, 2 a t)$ a point on $y^{2} = 4 a x$
Point M is $(- a, 2 a t)$
so equation of line passing through PA is
$y - 0 = \frac{2 a t}{a t^{2}} (x - 0)$
$\Rightarrow t y = 2 x$
it intersect $x = - a$ at $(- a, - \frac{2 a}{t})$
MD is the diameter of the circle, where vertex are $M (- a, 2 a t)$ & $D (- a, \frac{- 2 a}{t})$
then equation of circle
$(x + a) (x + a) + (y - 2 a t) (y + \frac{2 a}{t}) = 0$
$(x + a)^{2} + y^{2} + y (\frac{2 a}{t} - 2 a t) - (2 a)^{2} = 0$
at x-axis, $y = 0$
$\Rightarrow (x + a)^{2} - (2 a)^{2} = 0$
$\Rightarrow x + a = \pm 2 a$
$x = - 3 a$ or (a)
Hence circle cut x-axis at $(- 3 a, 0)$ and $(a, 0)$ .

Suggest Corrections

Similar questions

Q. P is any point on the parabola

y^{2} = 4 a x

whose vertex is A. PA is produced to meet the directrix in D & M is the foot of the perpendicular from P on the directrix. The angle subtended by MD at the focus is:

Q. The tangent PT and the normal PN to the parabola

y^{2} = 4 a x

at a point P on it meet its axis at points T and N, respectively. The locus of the centroid of the triangle PTN is a parabola, whose

PQ is a double ordinate of the parabola $y^{2}$ = 4ax. If the normal at a point P intersects the line passing through Q and parallel to axis of the parabola at G. Then locus of G is a parabola with

Q. If the normals at P and Q meet on the parabola, prove that the point of intersection of the tangents at P and Q lies either on a certain straight line, which is parallel to the tangent at the vertex, or on the curve whose equation is

y^{2} (x + 2 a) + 4 a^{3} = 0

Q. Assertion :Length of the common chord of the parabola

y^{2} = 8 x

and the circle

x^{2} + y^{2} = 9

is less than the length of the latus rectum of the parabola. Reason: If the vertex of a parabola lies at the point

(a, 0)

and the directrix is

x + a = 0

, then the focus of the parabola is at the point

(2 a, 0)

Join BYJU'S Learning Program

P is a point on the parabola y2=4ax(a>0) whose vertex is A. PA is produced to meet the directrix in D and M is the foot of the perpendicular from P on the directrix. If a circle is described on MD as a dimeter then it intersects the x-axis at a point whose co-ordinates are:

P is a point on the parabola $y^{2} = 4 a x (a > 0)$ whose vertex is A. PA is produced to meet the directrix in D and M is the foot of the perpendicular from P on the directrix. If a circle is described on MD as a dimeter then it intersects the x-axis at a point whose co-ordinates are: