JEE Main · 2019 · 09 Apr 2019 · Shift 2 · Circle

The common tangent to the circles

x^{2} + y^{2} = 4

and

x^{2} + y^{2} + 6 x + 8 y - 24 = 0

also passes through the point

(A) $(- 4, 6)$
(B) $(6, - 2)$ ✓ Correct
(C) $(- 6, 4)$
(D) $(4, - 2)$

Answer: (B)

Solution

The first circle,

x^{2} + y^{2} = 4

, has centre

(0, 0)

and radius

2

. The second,

x^{2} + y^{2} + 6 x + 8 y - 24 = 0

, has centre

(- 3, - 4)

and radius

9 + 16 + 24 = 7

. The distance between the centres is

9 + 16 = 5

, which equals

7 - 2

, so the circles touch internally and share one common tangent at the point of contact. That tangent is found from

S_{1} - S_{2} = 0

(x^{2} + y^{2} - 4) - (x^{2} + y^{2} + 6 x + 8 y - 24) = 0

gives

- 6 x - 8 y + 20 = 0

, or

3 x + 4 y - 10 = 0

. Testing

(6, - 2)

3 (6) + 4 (- 2) - 10 = 18 - 8 - 10 = 0

, so the tangent passes through

(6, - 2)

JEE Main · 2019 · 09 Apr 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 →