Let S be a square of unit area....

MathematicsDistance and Section FormulasJEE Advanced 1997Moderate

Let $S$ be a square of unit area. Consider any quadrilateral which has one vertex on each side of $S$ . If $a, b, c$ , and $d$ denote the lengths of the sides of the quadrilateral, prove that $2 \leq a^{2} + b^{2} + c^{2} + d^{2} \leq 4$ .

Visualized Solution (English)

Visualizing the Problem

Let the square $S$ have vertices at $(0, 0), (1, 0), (1, 1)$ , and $(0, 1)$ .
A quadrilateral is inscribed such that each vertex lies on one side of $S$ .
We need to prove $2 \leq a^{2} + b^{2} + c^{2} + d^{2} \leq 4$ .

Assigning Coordinates

Let the vertices of the quadrilateral be:
Vertex $P$ on $y = 0$ : $(x, 0)$
Vertex $Q$ on $x = 1$ : $(1, y)$
Vertex $R$ on $y = 1$ : $(z, 1)$
Vertex $S$ on $x = 0$ : $(0, w)$
Note that $0 \leq x, y, z, w \leq 1$ .

Applying the Distance Formula

Using the distance formula $d^{2} = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$ :
$a^{2} = (1 - x)^{2} + y^{2}$
$b^{2} = (1 - y)^{2} + (1 - z)^{2}$
$c^{2} = z^{2} + (1 - w)^{2}$
$d^{2} = x^{2} + w^{2}$

Summing the Squares

Sum $L = a^{2} + b^{2} + c^{2} + d^{2}$
$L = (1 - 2 x + x^{2} + y^{2}) + (1 - 2 y + y^{2} + 1 - 2 z + z^{2}) + (z^{2} + 1 - 2 w + w^{2}) + (x^{2} + w^{2})$
$L = 2 x^{2} - 2 x + 2 y^{2} - 2 y + 2 z^{2} - 2 z + 2 w^{2} - 2 w + 4$

Completing the Square

Rewrite $L$ by completing the square for each variable:
$2 x^{2} - 2 x + 1 = 2 (x^{2} - x + \frac{1}{4}) + \frac{1}{2} = 2 (x - \frac{1}{2})^{2} + \frac{1}{2}$
Applying this to all variables:
$L = 2 [(x - \frac{1}{2})^{2} + (y - \frac{1}{2})^{2} + (z - \frac{1}{2})^{2} + (w - \frac{1}{2})^{2}] + 2$

Finding the Minimum and Maximum

Since $(v a r - \frac{1}{2})^{2} \geq 0$ , the minimum value is $L_{min} = 2 (0) + 2 = 2$ .
This occurs when $x = y = z = w = \frac{1}{2}$ (midpoints).
Since $0 \leq v a r \leq 1$ , the maximum value of $(v a r - \frac{1}{2})^{2}$ is $\frac{1}{4}$ (at $0$ or $1$ ).
The maximum value is $L_{ma x} = 2 (\frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4}) + 2 = 2 (1) + 2 = 4$ .
Therefore, $2 \leq a^{2} + b^{2} + c^{2} + d^{2} \leq 4$ .

Key Takeaways

Key Takeaway: Coordinate geometry simplifies geometric constraints into manageable algebraic functions.
Optimization: The sum of squares of distances is minimized at the midpoints and maximized at the vertices of the bounding square.
Next Challenge: Try proving a similar bound for a general rectangle with sides $L$ and $W$ .

00:00 / 00:00

Conceptually Similar Problems

MathematicsVector (Cross) ProductJEE Advanced 1987Moderate

View in:English Hindi

If $A, B, C, D$ are any four points in space, prove that $∣ A B \times C D + B C \times A D + C A \times B D ∣ = 4 (area of triangle A B C)$ .

MathematicsProperties of TrianglesJEE Advanced 2001Moderate

View in:English Hindi

If $Δ$ is the area of a triangle with side lengths $a, b, c$ , then show that $Δ \leq \frac{1}{4} (a + b + c) ab c$ . Also show that the equality occurs in the above inequality if and only if $a = b = c$ .

MathematicsArea Bounded by CurvesJEE Advanced 1995Difficult

View in:English Hindi

Consider a square with vertices at $(1, 1), (- 1, 1), (- 1, - 1)$ and $(1, - 1)$ . Let $S$ be the region consisting of all points inside the square which are nearer to the origin than to any edge. Sketch the region $S$ and find its area.

Answer:

\frac{16 2 - 20}{3}

MathematicsScalar (Dot) ProductJEE Advanced 2001SEasy

View in:English Hindi

If $a, b$ and $c$ are unit vectors, then $∣ a - b ∣^{2} + ∣ b - c ∣^{2} + ∣ c - a ∣^{2}$ does NOT exceed

(A)

4

(B)

9

(C)

8

(D)

6

Answer:B

MathematicsGeometric Progression (G.P.)JEE Advanced 1999Easy

View in:English Hindi

Let $S_{1}, S_{2}, \dots$ be squares such that for each $n \geq 1$ , the length of a side of $S_{n}$ equals the length of a diagonal of $S_{n + 1}$ . If the length of a side of $S_{1}$ is 10 cm, then for which of the following values of $n$ is the area of $S_{n}$ less than 1 sq. cm?

* Multiple Correct Options

(A)

(B)

(C)

(D)

Answer:B, C, D

MathematicsProperties of TrianglesJEE Advanced 1991Moderate

View in:English Hindi

In a triangle of base $a$ the ratio of the other two sides is $r (< 1)$ . Show that the altitude of the triangle is less than or equal to $\frac{a r}{1 - r ^{2}}$ .

MathematicsConditional IdentitiesJEE Advanced 2000Easy

View in:English Hindi

In any triangle $A B C$ , prove that $cot \frac{A}{2} + cot \frac{B}{2} + cot \frac{C}{2} = cot \frac{A}{2} cot \frac{B}{2} cot \frac{C}{2}$ .

MathematicsEquation of Tangent and NormalJEE Advanced 2007Moderate

View in:English Hindi

Let $A B C D$ be a quadrilateral with area 18, with side $A B$ parallel to the side $C D$ and $2 A B = C D$ . Let $A D$ be perpendicular to $A B$ and $C D$ . If a circle is drawn inside the quadrilateral $A B C D$ touching all the sides, then its radius is

(A)

(B)

(C)

3/2

(D)

Answer:B

MathematicsArithmetic Progression (A.P.)JEE Advanced 2000Easy

View in:English Hindi

The fourth power of the common difference of an arithmetic progression with integer entries is added to the product of any four consecutive terms of it. Prove that the resulting sum is the square of an integer.

MathematicsStandard Equations of Parabola, Ellipse, and HyperbolaJEE Advanced 2006Difficult

View in:English Hindi

Comprehension Passage

A B C D

is a square of side length 2 units.

C_{1}

is the circle touching all the sides of the square

A B C D

and

C_{2}

is the circumcircle of square

A B C D

L

is a fixed line in the same plane and

R

is a fixed point.

Question 1:

If $P$ is any point of $C_{1}$ and $Q$ is another point on $C_{2}$ , then $\frac{P A ^{2} + P B ^{2} + P C ^{2} + P D ^{2}}{Q A ^{2} + Q B ^{2} + Q C ^{2} + Q D ^{2}}$ is equal to

(A)

0.75

(B)

1.25

(C)

(D)

0.5

Answer:A

Question 2:

If a circle is such that it touches the line $L$ and the circle $C_{1}$ externally, such that both the circles are on the same side of the line, then the locus of centre of the circle is

(A)

ellipse

(B)

hyperbola

(C)

parabola

(D)

pair of straight line

Answer:C

Question 3:

A line $L^{'}$ through $A$ is drawn parallel to $B D$ . Point $S$ moves such that its distances from the line $B D$ and the vertex $A$ are equal. If locus of $S$ cuts $L^{'}$ at $T_{2}$ and $T_{3}$ and $A C$ at $T_{1}$ , then area of $Δ T_{1} T_{2} T_{3}$ is

(A)

1/2 sq. units

(B)

2/3 sq. units

(C)

1 sq. units

(D)

2 sq. units

Answer:C

Visualized Solution (English)

Visualizing the Problem

Assigning Coordinates

Applying the Distance Formula

Summing the Squares

Completing the Square

Finding the Minimum and Maximum

Key Takeaways

Conceptually Similar Problems

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If A,B,C,D are any four points in space, prove that ∣AB×CD+BC×AD+CA×BD∣=4(area of triangle ABC).

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a,b and c are unit vectors, then ∣a−b∣2+∣b−c∣2+∣c−a∣2 does NOT exceed

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } In a triangle of base a the ratio of the other two sides is r(<1). Show that the altitude of the triangle is less than or equal to 1−r2ar​.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } In any triangle ABC, prove that cot2A​+cot2B​+cot2C​=cot2A​cot2B​cot2C​.

The fourth power of the common difference of an arithmetic progression with integer entries is added to the product of any four consecutive terms of it. Prove that the resulting sum is the square of an integer.

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If P is any point of C1​ and Q is another point on C2​, then QA2+QB2+QC2+QD2PA2+PB2+PC2+PD2​ is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If A,B,C,D are any four points in space, prove that ∣AB×CD+BC×AD+CA×BD∣=4(area of triangle ABC).

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a,b and c are unit vectors, then ∣a−b∣2+∣b−c∣2+∣c−a∣2 does NOT exceed

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } In a triangle of base a the ratio of the other two sides is r(<1). Show that the altitude of the triangle is less than or equal to 1−r2ar​.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } In any triangle ABC, prove that cot2A​+cot2B​+cot2C​=cot2A​cot2B​cot2C​.

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If P is any point of C1​ and Q is another point on C2​, then QA2+QB2+QC2+QD2PA2+PB2+PC2+PD2​ is equal to

If $A, B, C, D$ are any four points in space, prove that $∣ A B \times C D + B C \times A D + C A \times B D ∣ = 4 (area of triangle A B C)$ .

If $a, b$ and $c$ are unit vectors, then $∣ a - b ∣^{2} + ∣ b - c ∣^{2} + ∣ c - a ∣^{2}$ does NOT exceed

In a triangle of base $a$ the ratio of the other two sides is $r (< 1)$ . Show that the altitude of the triangle is less than or equal to $\frac{a r}{1 - r ^{2}}$ .

In any triangle $A B C$ , prove that $cot \frac{A}{2} + cot \frac{B}{2} + cot \frac{C}{2} = cot \frac{A}{2} cot \frac{B}{2} cot \frac{C}{2}$ .

If $P$ is any point of $C_{1}$ and $Q$ is another point on $C_{2}$ , then $\frac{P A ^{2} + P B ^{2} + P C ^{2} + P D ^{2}}{Q A ^{2} + Q B ^{2} + Q C ^{2} + Q D ^{2}}$ is equal to

If $A, B, C, D$ are any four points in space, prove that $∣ A B \times C D + B C \times A D + C A \times B D ∣ = 4 (area of triangle A B C)$ .

If $a, b$ and $c$ are unit vectors, then $∣ a - b ∣^{2} + ∣ b - c ∣^{2} + ∣ c - a ∣^{2}$ does NOT exceed

In a triangle of base $a$ the ratio of the other two sides is $r (< 1)$ . Show that the altitude of the triangle is less than or equal to $\frac{a r}{1 - r ^{2}}$ .

In any triangle $A B C$ , prove that $cot \frac{A}{2} + cot \frac{B}{2} + cot \frac{C}{2} = cot \frac{A}{2} cot \frac{B}{2} cot \frac{C}{2}$ .

If $P$ is any point of $C_{1}$ and $Q$ is another point on $C_{2}$ , then $\frac{P A ^{2} + P B ^{2} + P C ^{2} + P D ^{2}}{Q A ^{2} + Q B ^{2} + Q C ^{2} + Q D ^{2}}$ is equal to