JEE Main · 2022 · 27 Jun 2022 · Shift 1 · Circle

A rectangle R with end points of one of its sides as

(1, 2)

and

(3, 6)

is inscribed in a circle. If the equation of a diameter of the circle is

2 x - y + 4 = 0

, then the area of R is ______.

Answer: 16

Solution

The side AB has endpoints

(1, 2)

and

(3, 6)

. The equation of line AB is:

\frac{y - 2}{x - 1} = \frac{6 - 2}{3 - 1} = 2 \Rightarrow y = 2 x

The given diameter has equation

2 x - y + 4 = 0

, or

y = 2 x + 4

, which has slope 2.

Since the diameter is parallel to AB (both have slope 2), the perpendicular distance between them gives half the width of the rectangle:

d = \frac{∣4∣}{2 ^{2} + 1 ^{2}} = \frac{4}{5}

So the width

B C = 2 d = \frac{8}{5}

.

The length

A B = (3 - 1)^{2} + (6 - 2)^{2} = 4 + 16 = 25

.

Area

= A B \times B C = 25 \times \frac{8}{5} = 16

JEE Main · 2022 · 27 Jun 2022 · Shift 1 · 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 →