JEE Main · 2022 · 24 Jun 2022 · Shift 2 · Circle

Let a circle

C : (x - h)^{2} + (y - k)^{2} = r^{2}, k > 0

, touch the x-axis at

(1, 0)

. If the line

x + y = 0

intersects the circle C at P and Q such that the length of the chord PQ is 2, then the value of

h + k + r

is equal to ____.

Answer: 7

Solution

Since the circle touches the x-axis at

(1, 0)

, we have

h = 1

and

k = r

(the center is at

(1, r)

with radius

r

).

The perpendicular distance from center

(1, r)

to the line

x + y = 0

is:

d = \frac{∣1 + r ∣}{2} = \frac{1 + r}{2}

Using the chord-distance formula with chord length

P Q = 2

r^{2} = d^{2} + 1^{2} = \frac{( 1 + r ) ^{2}}{2} + 1

2 r^{2} = (1 + r)^{2} + 2 = 1 + 2 r + r^{2} + 2

r^{2} - 2 r - 3 = 0 \Rightarrow (r - 3) (r + 1) = 0

Since

r > 0

, we have

r = 3

.

Therefore,

h + k + r = 1 + 3 + 3 = 7

JEE Main · 2022 · 24 Jun 2022 · Shift 2 · Integer answer

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 →