Distance and Section Formulas

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JEE Main 2012

If the line $2 x + y = k$ passes through the point which divides the line segment joining the points $(1, 1)$ and $(2, 4)$ in the ratio $3 : 2$ , then $k$ equals

\frac{29}{5}

(B)

(C)

(D)

\frac{11}{5}

JEE Main 2003

LEVELBoard

If the equation of the locus of a point equidistant from the point $(a_{1}, b_{1})$ and $(a_{2}, b_{2})$ is $(a_{1} - a_{2}) x + (b_{1} - b_{2}) y + c = 0$ , then the value of 'c' is

(A)

a_{1}^{2} + b_{1}^{2} - a_{2}^{2} - b_{2}^{2}

(B)

\frac{1}{2} (a_{2}^{2} + b_{2}^{2} - a_{1}^{2} - b_{1}^{2})

(C)

a_{1}^{2} - a_{2}^{2} + b_{1}^{2} - b_{2}^{2}

(D)

\frac{1}{2} (a_{1}^{2} + a_{2}^{2} + b_{1}^{2} + b_{2}^{2})

JEE Main 2002

LEVELBoard

A triangle with vertices $(4, 0), (- 1, - 1), (3, 5)$ is

(A)

isosceles and right angled

(B)

isosceles but not right angled

(C)

right angled but not isosceles

(D)

neither right angled nor isosceles

JEE Advanced 2013

LEVELJEE Main

For $a > b > c > 0$ , the distance between $(1, 1)$ and the point of intersection of the lines $a x + b y + c = 0$ and $b x + a y + c = 0$ is less than $22$ . Then

* Multiple Correct Options

(A)

a + b - c > 0

(B)

a - b + c < 0

(C)

a - b + c > 0

(D)

a + b - c < 0

JEE Advanced 1998

LEVELBoard

If $(1, 2), (4, 6), (5, 7)$ and $(a, b)$ are the vertices of a parallelogram $P QR S$ , then

(A)

a = 2, b = 4

(B)

a = 3, b = 4

(C)

a = 2, b = 3

(D)

a = 3, b = 5

JEE Advanced 1980

LEVELJEE Main

The point $(4, 1)$ undergoes the following three transformations successively. \n(i) Reflection about the line $y = x$ . \n(ii) Translation through a distance 2 units along the positive direction of x-axis. \n(iii) Rotation through an angle $π /4$ about the origin in the counter clockwise direction. \nThen the final position of the point is given by the coordinates.

(A)

(\frac{1}{2}, \frac{7}{2})

(B)

(- 2, 72)

(C)

(- \frac{1}{2}, \frac{7}{2})

(D)

(2, 72)

JEE Advanced 1978

LEVELJEE Main

A straight line segment of length $l$ moves with its ends on two mutually perpendicular lines. Find the locus of the point which divides the line segment in the ratio $1 : 2$ .

JEE Advanced 1997

LEVELJEE Main

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 $2 \leq a^{2} + b^{2} + c^{2} + d^{2} \leq 4$ .

JEE Main 2021 (27 July Shift 2)

LEVELJEE Main

The point $P (a, b)$ undergoes the following three transformations successively: (a) reflection about the line $y = x$ . (b) translation through 2 units along the positive direction of x-axis. (c) rotation through angle $\frac{π}{4}$ about the origin in the anti-clockwise direction. If the co-ordinates of the final position of the point $P$ are $(- \frac{1}{2}, \frac{7}{2})$ , then the value of $2 a + b$ is equal to :

JEE Main 2024 (31 Jan Shift 2)

LEVELJEE Main

Let $A (- 2, - 1), B (1, 0), C (α, β)$ and $D (γ, δ)$ be the vertices of a parallelogram $A B C D$ . If the point $C$ lies on $2 x - y = 5$ and the point $D$ lies on $3 x - 2 y = 6$ , then the value of $∣ α + β + γ + δ ∣$ is equal to _______.

JEE Main 2025 (January)

LEVELJEE Advanced

A rod of length eight units moves such that its ends A and B always lie on the lines $x - y + 2 = 0$ and $y + 2 = 0$ respectively. If the locus of the point P, that divides the rod AB internally in the ratio 2:1 is $9 (x^{2} + α y^{2} + β x y + γ x + 28 y) - 76 = 0$ , then $α - β - γ$ is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the line 2x+y=k passes through the point which divides the line segment joining the points (1,1) and (2,4) in the ratio 3:2, then k equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the equation of the locus of a point equidistant from the point (a1​,b1​) and (a2​,b2​) is (a1​−a2​)x+(b1​−b2​)y+c=0, then the value of 'c' is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A triangle with vertices (4,0),(−1,−1),(3,5) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For a>b>c>0, the distance between (1,1) and the point of intersection of the lines ax+by+c=0 and bx+ay+c=0 is less than 22​. Then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If (1,2),(4,6),(5,7) and (a,b) are the vertices of a parallelogram PQRS, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A straight line segment of length l moves with its ends on two mutually perpendicular lines. Find the locus of the point which divides the line segment in the ratio 1:2.

If the line $2 x + y = k$ passes through the point which divides the line segment joining the points $(1, 1)$ and $(2, 4)$ in the ratio $3 : 2$ , then $k$ equals

If the equation of the locus of a point equidistant from the point $(a_{1}, b_{1})$ and $(a_{2}, b_{2})$ is $(a_{1} - a_{2}) x + (b_{1} - b_{2}) y + c = 0$ , then the value of 'c' is

A triangle with vertices $(4, 0), (- 1, - 1), (3, 5)$ is

For $a > b > c > 0$ , the distance between $(1, 1)$ and the point of intersection of the lines $a x + b y + c = 0$ and $b x + a y + c = 0$ is less than $22$ . Then

If $(1, 2), (4, 6), (5, 7)$ and $(a, b)$ are the vertices of a parallelogram $P QR S$ , then

A straight line segment of length $l$ moves with its ends on two mutually perpendicular lines. Find the locus of the point which divides the line segment in the ratio $1 : 2$ .