For points P = (x1, y1) and Q = (x2,...

MathematicsVarious Forms of Equations of a LineJEE Advanced 2000Moderate

For points $P = (x_{1}, y_{1})$ and $Q = (x_{2}, y_{2})$ of the co-ordinate plane, a new distance $d (P, Q)$ is defined by $d (P, Q) = ∣ x_{1} - x_{2} ∣ + ∣ y_{1} - y_{2} ∣$ . Let $O = (0, 0)$ and $A = (3, 2)$ . Prove that the set of points in the first quadrant which are equidistant (with respect to the new distance) from $O$ and $A$ consists of the union of a line segment of finite length and an infinite ray. Sketch this set in a labelled diagram.

Visualized Solution (Hindi)

Introduction to Manhattan Distance

Given Points: $O = (0, 0)$ and $A = (3, 2)$
New Distance Definition: $d (P, Q) = ∣ x_{1} - x_{2} ∣ + ∣ y_{1} - y_{2} ∣$
Objective: Find the locus of $P (x, y)$ such that $d (P, O) = d (P, A)$ in the first quadrant.

Distance from Origin .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } d (P, O)

For $P (x, y)$ in the first quadrant, $x \geq 0$ and $y \geq 0$ .
The distance from origin $O (0, 0)$ is:
$d (P, O) = ∣ x - 0∣ + ∣ y - 0∣ = x + y$

Distance from Point .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A (3, 2)

The distance from point $A (3, 2)$ is:
$d (P, A) = ∣ x - 3∣ + ∣ y - 2∣$

The Equidistant Condition

Equating the distances:
$x + y = ∣ x - 3∣ + ∣ y - 2∣$

Case 1: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } y \geq 2

Assume $y \geq 2$ and $x < 3$ :
$x + y = (3 - x) + (y - 2)$
$x + y = 1 - x + y$
$2 x = 1 \Rightarrow x = \frac{1}{2}$
This forms an infinite ray starting from $(\frac{1}{2}, 2)$ upwards.

Case 2: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } y < 2

Assume $y < 2$ and $x < 3$ :
$x + y = (3 - x) + (2 - y)$
$x + y = 5 - x - y$
$2 x + 2 y = 5 \Rightarrow x + y = \frac{5}{2}$
This forms a finite line segment from $(\frac{5}{2}, 0)$ to $(\frac{1}{2}, 2)$ .

Final Locus and Summary

Final Set: Union of the segment $x + y = \frac{5}{2}$ for $y \in [0, 2]$ and the ray $x = \frac{1}{2}$ for $y \geq 2$ .
Key Takeaway: Manhattan distance results in piecewise linear loci.
Next Challenge: What is the shape of a 'circle' defined by $d (P, O) = r$ ?

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Conceptually Similar Problems

MathematicsStandard and General Equation of a CircleJEE Advanced 1987Moderate

View in:English Hindi

Let a given line $L_{1}$ intersects the $x$ and $y$ axes at $P$ and $Q$ , respectively. Let another line $L_{2}$ , perpendicular to $L_{1}$ , cut the $x$ and $y$ axes at $R$ and $S$ , respectively. Show that the locus of the point of intersection of the lines $P S$ and $QR$ is a circle passing through the origin.

MathematicsVarious Forms of Equations of a LineJEE Advanced 2002Moderate

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A straight line $L$ through the origin meets the lines $x + y = 1$ and $x + y = 3$ at $P$ and $Q$ respectively. Through $P$ and $Q$ two straight lines $L_{1}$ and $L_{2}$ are drawn, parallel to $2 x - y = 5$ and $3 x + y = 5$ respectively. Lines $L_{1}$ and $L_{2}$ intersect at $R$ . Show that the locus of $R$ , as $L$ varies, is a straight line.

MathematicsFamily of LinesJEE Advanced 1991Easy

View in:English Hindi

Let the algebraic sum of the perpendicular distances from the points $(2, 0), (0, 2)$ and $(1, 1)$ to a variable straight line be zero; then the line passes through a fixed point whose coordinates are .........

Answer:

(1, 1)

MathematicsStandard and General Equation of a CircleJEE Advanced 1997Moderate

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Let $C$ be any circle with centre $(0, 2)$ . Prove that at the most two rational points can be there on $C$ . (A rational point is a point both of whose coordinates are rational numbers.)

MathematicsEquation of a PlaneJEE Advanced 2004Moderate

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$P_{1}$ and $P_{2}$ are planes passing through origin. $L_{1}$ and $L_{2}$ are two line on $P_{1}$ and $P_{2}$ respectively such that their intersection is origin. Show that there exists points $A, B, C$ whose permutation $A^{'}, B^{'}, C^{'}$ can be chosen such that (i) $A$ is on $L_{1}, B$ on $P_{1}$ but not on $L_{1}$ and $C$ not on $P_{1}$ (ii) $A^{'}$ is on $L_{2}, B^{'}$ on $P_{2}$ but not on $L_{2}$ and $C^{'}$ not on $P_{2}$ .

MathematicsDistance of a Point from a LineJEE Advanced 2014Moderate

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For a point $P$ in the plane, let $d_{1} (P)$ and $d_{2} (P)$ be the distance of the point $P$ from the lines $x - y = 0$ and $x + y = 0$ respectively. The area of the region $R$ consisting of all points $P$ lying in the first quadrant of the plane and satisfying $2 \leq d_{1} (P) + d_{2} (P) \leq 4$ , is .........

Answer:6

MathematicsStandard and General Equation of a CircleJEE Advanced 1983Moderate

View in:English Hindi

Through a fixed point $(h, k)$ secants are drawn to the circle $x^{2} + y^{2} = r^{2}$ . Show that the locus of the mid-points of the secants intercepted by the circle is $x^{2} + y^{2} = h x + k y$ .

MathematicsVarious Forms of Equations of a LineJEE Advanced 1983Moderate

View in:English Hindi

The end $A, B$ of a straight line segment of constant length $c$ slide upon the fixed rectangular axes $O X, O Y$ respectively. If the rectangle $O A P B$ be completed, then show that the locus of the foot of the perpendicular drawn from $P$ to $A B$ is $x^{2/3} + y^{2/3} = c^{2/3}$ .

Answer:x^{2/3} + y^{2/3} = c^{2/3}

MathematicsEquation of Tangent and NormalJEE Advanced 2016Moderate

View in:English Hindi

Let $R S$ be the diameter of the circle $x^{2} + y^{2} = 1$ , where $S$ is the point $(1, 0)$ . Let $P$ be a variable point (other than $R$ and $S$ ) on the circle and tangents to the circle at $S$ and $P$ meet at the point $Q$ . The normal to the circle at $P$ intersects a line drawn through $Q$ parallel to $R S$ at point $E$ . Then the locus of $E$ passes through the point(s)

* Multiple Correct Options

(A)

(\frac{1}{3}, \frac{1}{3})

(B)

(\frac{1}{4}, \frac{1}{2})

(C)

(\frac{1}{3}, - \frac{1}{3})

(D)

(\frac{1}{4}, - \frac{1}{2})

Answer:A, C

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 (Hindi)

Introduction to Manhattan Distance

Distance from Origin $d (P, O)$

Distance from Point $A (3, 2)$

The Equidistant Condition

Case 1: $y \geq 2$

Case 2: $y < 2$

Final Locus and Summary

Conceptually Similar Problems

Let the algebraic sum of the perpendicular distances from the points $(2, 0), (0, 2)$ and $(1, 1)$ to a variable straight line be zero; then the line passes through a fixed point whose coordinates are .........

Let $C$ be any circle with centre $(0, 2)$ . Prove that at the most two rational points can be there on $C$ . (A rational point is a point both of whose coordinates are rational numbers.)

Through a fixed point $(h, k)$ secants are drawn to the circle $x^{2} + y^{2} = r^{2}$ . Show that the locus of the mid-points of the secants intercepted by the circle is $x^{2} + y^{2} = h x + k y$ .

The end $A, B$ of a straight line segment of constant length $c$ slide upon the fixed rectangular axes $O X, O Y$ respectively. If the rectangle $O A P B$ be completed, then show that the locus of the foot of the perpendicular drawn from $P$ to $A B$ is $x^{2/3} + y^{2/3} = c^{2/3}$ .

Comprehension Passage

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 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 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

Let the algebraic sum of the perpendicular distances from the points $(2, 0), (0, 2)$ and $(1, 1)$ to a variable straight line be zero; then the line passes through a fixed point whose coordinates are .........

Let $C$ be any circle with centre $(0, 2)$ . Prove that at the most two rational points can be there on $C$ . (A rational point is a point both of whose coordinates are rational numbers.)

Through a fixed point $(h, k)$ secants are drawn to the circle $x^{2} + y^{2} = r^{2}$ . Show that the locus of the mid-points of the secants intercepted by the circle is $x^{2} + y^{2} = h x + k y$ .

The end $A, B$ of a straight line segment of constant length $c$ slide upon the fixed rectangular axes $O X, O Y$ respectively. If the rectangle $O A P B$ be completed, then show that the locus of the foot of the perpendicular drawn from $P$ to $A B$ is $x^{2/3} + y^{2/3} = c^{2/3}$ .

Comprehension Passage

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 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 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

Visualized Solution (Hindi)

Introduction to Manhattan Distance

Distance from Origin .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } d(P,O)

Distance from Point .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A(3,2)

The Equidistant Condition

Case 1: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } y≥2

Case 2: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } y<2

Final Locus and Summary

Conceptually Similar Problems

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Through a fixed point (h,k) secants are drawn to the circle x2+y2=r2. Show that the locus of the mid-points of the secants intercepted by the circle is x2+y2=hx+ky.

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; } Through a fixed point (h,k) secants are drawn to the circle x2+y2=r2. Show that the locus of the mid-points of the secants intercepted by the circle is x2+y2=hx+ky.

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

Distance from Origin $d (P, O)$

Distance from Point $A (3, 2)$

Case 1: $y \geq 2$

Case 2: $y < 2$

Through a fixed point $(h, k)$ secants are drawn to the circle $x^{2} + y^{2} = r^{2}$ . Show that the locus of the mid-points of the secants intercepted by the circle is $x^{2} + y^{2} = h x + k y$ .

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

Through a fixed point $(h, k)$ secants are drawn to the circle $x^{2} + y^{2} = r^{2}$ . Show that the locus of the mid-points of the secants intercepted by the circle is $x^{2} + y^{2} = h x + k y$ .

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