JEE Main 2025 (January)

LEVELJEE Main

Animated Solution for Mathematics - Circles: Let the line $x + y = 1$ meet the circle $x^{2} + y^{2} = 4$ at the points A and B. If the line perpendicular to AB and passing through the mid point of the chord AB intersects the circle at C and D, then the area of the quadrilateral ADBC is equal to:

Q: How do I know for sure that the diagonals are perpendicular?

The problem explicitly states that the line $CD$ is perpendicular to the chord $AB$. Since $AB$ and $CD$ are the diagonals of the quadrilateral $ADBC$, and they intersect at a $90^{\circ}$ angle, the formula for the area of a kite-like quadrilateral applies perfectly.

Q: Is the quadrilateral $ADBC$ always a kite?

Yes! Because $CD$ is the perpendicular bisector of $AB$, the distance from $C$ to $A$ equals the distance from $C$ to $B$, and similarly for $D$. This symmetry makes it a kite, which is a specific type of quadrilateral where the area formula $\frac{1}{2} d_1 d_2$ is always valid.

Select Answer:

Visualized Solution

Circle Equation

Circle: $x^{2} + y^{2} = 4$
Center $O = (0, 0)$
Radius $r = 4 = 2$

The Chord $A B$

Line equation: $x + y = 1$
Intersects circle at points $A$ and $B$
Forms chord $A B$

Perpendicular Distance $p$

Drop a perpendicular from center $O (0, 0)$ to chord $A B$
Let the distance be $p$
Formula: $p = \frac{∣ a x _{1} + b y _{1} + c ∣}{a ^{2} + b ^{2}}$

Calculating $p$

Line: $x + y - 1 = 0$
Point: $(0, 0)$
$p = \frac{∣0 + 0 - 1∣}{1 ^{2} + 1 ^{2}}$
$p = \frac{1}{2}$

Length of Chord $A B$

Formula: $A B = 2 r^{2} - p^{2}$
$A B = 2 2^{2} - (\frac{1}{2})^{2}$
$A B = 2 4 - \frac{1}{2} = 2 \frac{7}{2} = 14$

Perpendicular Bisector $C D$

Line $C D$ is perpendicular to $A B$
Passes through the midpoint of $A B$
Therefore, $C D$ is the perpendicular bisector of $A B$

$C D$ as a Diameter

Property: The perpendicular bisector of a chord always passes through the center.
Since $C D$ passes through $(0, 0)$ , it is a diameter.
Length $C D = 2 r = 2 \times 2 = 4$

Quadrilateral $A D B C$

Vertices: $A, D, B, C$
Diagonals: $A B$ and $C D$
The diagonals are perpendicular to each other.

Area of Quadrilateral

For a quadrilateral with perpendicular diagonals $d_{1}$ and $d_{2}$ :
Area $= \frac{1}{2} \times d_{1} \times d_{2}$
Here, $d_{1} = A B$ and $d_{2} = C D$

Final Area Calculation

Area $= \frac{1}{2} \times A B \times C D$
Area $= \frac{1}{2} \times 14 \times 4$
Area $= 214$

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The Sigma Insight: Standard and General Equation of a Circle

Solution Diagram

Analyzing the Setup

The circle is defined by the equation

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

. This identifies a circle centered at the origin

(0, 0)

with a radius

r = 2

The line

x + y = 1

creates a chord

A B

. We are tasked with finding the area of the quadrilateral

A D B C

, where

C D

is the perpendicular bisector of

A B

passing through the circle.

The Anatomy of the Chord

To find the length of the chord

A B

, we calculate the perpendicular distance

p

from the center

(0, 0)

to the line

x + y - 1 = 0

. Using the distance formula:

p = \frac{∣0 + 0 - 1∣}{1 ^{2} + 1 ^{2}} = \frac{1}{2}

We utilize the geometric relationship between the radius

r

, the distance

p

, and the chord length

L

. By the Pythagorean theorem, the half-length of the chord is

r^{2} - p^{2}

The full length of the chord

A B

is given by:

A B = 2 r^{2} - p^{2} = 2 2^{2} - (\frac{1}{2})^{2}

A B = 2 4 - \frac{1}{2} = 2 \frac{7}{2} = 14

The Power of the Perpendicular Bisector

The line

C D

is defined as the perpendicular bisector of the chord

A B

. A fundamental property of circles is that the perpendicular bisector of any chord must pass through the center of the circle.

Since

C D

passes through the center and its endpoints lie on the circle,

C D

is a diameter of the circle. Given the radius

r = 2

, the length of the second diagonal is:

C D = 2 r = 2 (2) = 4

Final Calculation

The quadrilateral

A D B C

has perpendicular diagonals

A B

and

C D

. The area of any quadrilateral with perpendicular diagonals is calculated as:

Area = \frac{1}{2} \times d_{1} \times d_{2}

Substituting our calculated lengths

d_{1} = 14

and

d_{2} = 4

Area = \frac{1}{2} \times 14 \times 4

Area

= 214

Common Traps & FAQs

Wait, why is the line

C D

automatically a diameter?

That is a fantastic observation! In any circle, the perpendicular bisector of a chord must pass through the center of the circle. Since the line

C D

is defined as the perpendicular bisector of chord

A B

, it is forced to pass through the origin

(0, 0)

. Any chord passing through the center is, by definition, a diameter.

Can I just find the coordinates of points A, B, C, and D and use the shoelace formula?

You absolutely could, and it would work! However, in the JEE, time is your most precious currency. Using the property that the area of a quadrilateral with perpendicular diagonals is

\frac{1}{2} d_{1} d_{2}

saves you from solving messy quadratic equations and dealing with potentially irrational coordinates.

How do I know for sure that the diagonals are perpendicular?

The problem explicitly states that the line

C D

is perpendicular to the chord

A B

. Since

A B

and

C D

are the diagonals of the quadrilateral

A D B C

, and they intersect at a

9 0^{\circ}

angle, the formula for the area of a kite-like quadrilateral applies perfectly.

Is the quadrilateral

A D B C

always a kite?

Yes! Because

C D

is the perpendicular bisector of

A B

, the distance from

C

A

equals the distance from

C

B

, and similarly for

D

. This symmetry makes it a kite, which is a specific type of quadrilateral where the area formula

\frac{1}{2} d_{1} d_{2}

is always valid.

What if the line

x + y = 1

didn't intersect the circle?

If the distance from the center to the line were greater than the radius (

p > r

), the line would not intersect the circle, and the chord

A B

would not exist. Always check the condition

p < r

first to ensure the geometry is physically possible!

Animated Solution for Mathematics - Circles: Let the line $x + y = 1$ meet the circle $x^{2} + y^{2} = 4$ at the points A and B. If the line perpendicular to AB and passing through the mid point of the chord AB intersects the circle at C and D, then the area of the quadrilateral ADBC is equal to:

Select Answer:

Visualized Solution

Circle Equation

The Chord $A B$

Perpendicular Distance $p$

Calculating $p$

Length of Chord $A B$

Perpendicular Bisector $C D$

$C D$ as a Diameter

Quadrilateral $A D B C$

Area of Quadrilateral

Final Area Calculation

The Sigma Insight: Standard and General Equation of a Circle

Analyzing the Setup

The Anatomy of the Chord

The Power of the Perpendicular Bisector

Final Calculation

Similar Questions

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A rectangle $R$ with end points of the 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 _______.

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 such that $(α - β) = \frac{π}{2}$ . Then the point of intersection of $A Q$ and $B P$ lies on :

Select Answer:

Visualized Solution

Circle Equation

The Chord .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } AB

Perpendicular Distance .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } p

Calculating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } p

Length of Chord .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } AB

Perpendicular Bisector .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } CD

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } CD as a Diameter

Quadrilateral .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ADBC

Area of Quadrilateral

Final Area Calculation

The Sigma Insight: Standard and General Equation of a Circle

Analyzing the Setup

The Anatomy of the Chord

The Power of the Perpendicular Bisector

Final Calculation

Similar Questions

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Perpendicular Distance $p$

Calculating $p$

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