JEE Main · 2019 · 11 Jan 2019 · Shift 2 · Circle

A circle cuts a chord of length 4a on the

x

-axis and passes through a point on the

y

-axis, distant 2b from the origin. Then the locus of the center of this circle, is:

(A)a hyperbola
(B)an ellipse
(C)a straight line
(D)a parabola✓ Correct

Answer: (D)

Solution

Let the centre be

(h, k)

with radius

r

. The chord cut on the

x

-axis has length

4 a

, so its half-length

2 a

and the distance

k

from the centre to the axis satisfy

k^{2} + (2 a)^{2} = r^{2}

. The circle passes through

(0, 2 b)

, so

h^{2} + (k - 2 b)^{2} = r^{2}

. Equating,

h^{2} + (k - 2 b)^{2} = k^{2} + 4 a^{2}

, which reduces to

h^{2} = 4 bk - 4 b^{2} + 4 a^{2}

. Replacing

(h, k)

(x, y)

, the locus is

x^{2} = 4 (b y - b^{2} + a^{2})

, a parabola.

JEE Main · 2019 · 11 Jan 2019 · Shift 2 · Single correct option

Practise more: all JEE Main PYQs on Circle →

More JEE Main questions on Circle

Let PQ and MN be two straight lines touching the circle $x^{2} + y^{2} - 4 x - 6 y - 3 = 0$ at the points A and B respectively. Let O be the centre of the circle and $∠ AOB = π /3$ . Then the locus of the point of intersection of the lines PQ and MN is:
JEE Main 2026View solution →
If P is a point on the circle $x^{2} + y^{2} = 4$ , Q is a point on the straight line $5 x + y + 2 = 0$ and $x - y + 1 = 0$ is the perpendicular bisector of PQ, then 13 times the sum of abscissa of all such point P is ______.
JEE Main 2026View solution →
Let the set of all values of r, for which the circles $(x + 1)^{2} + (y + 4)^{2} = r^{2}$ and $x^{2} + y^{2} - 4 x - 2 y - 4 = 0$ intersect at two distinct points be the interval ( $α$ , $β$ ). Then $α β$ is equal to
JEE Main 2026View solution →
Let a circle of radius 4 pass through the origin O , the points $A (- 3 a, 0)$ and $B (0, - 2 b)$ , where a and b are real parameters and $ab \neq = 0$ . Then the locus of the centroid of $△ OAB$ is a cir …
JEE Main 2026View solution →
Let $y = x$ be the equation of a chord of the circle $C_{1}$ (in the closed half-plane $x \geq 0$ ) of diameter 10 passing through the origin. Let $C_{2}$ be another circle described on the given chord as its diameter. If the equation of the chord …
JEE Main 2026View solution →
Let the circle $x^{2} + y^{2} = 4$ intersect $x$ -axis at the points $A (a, 0), a > 0$ and $B (b, 0)$ . Let $P (2 cos α, 2 sin α), 0 < α < \frac{π}{2}$ and $Q (2 cos β, 2 sin β)$ be two points …
JEE Main 2026View solution →