Centroid, Incenter, Orthocenter, and Circumcenter

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JEE Main 2013

The x-coordinate of the incentre of the triangle that has the coordinates of mid points of its sides as $(0, 1), (1, 1)$ and $(1, 0)$ is

2 + 2

(B)

2 - 2

(C)

1 + 2

(D)

1 - 2

JEE Main 2005

LEVELBoard

If a vertex of a triangle is $(1, 1)$ and the mid points of two sides through this vertex are $(- 1, 2)$ and $(3, 2)$ then the centroid of the triangle is

(A)

(- 1, \frac{7}{3})

(B)

(\frac{- 1}{3}, \frac{7}{3})

(C)

(1, \frac{7}{3})

(D)

(\frac{1}{3}, \frac{7}{3})

JEE Main 2004

LEVELJEE Main

Let $A (2, - 3)$ and $B (- 2, 3)$ be vertices of a triangle $A B C$ . If the centroid of this triangle moves on the line $2 x + 3 y = 1$ , then the locus of the vertex $C$ is the line

(A)

3 x - 2 y = 3

(B)

2 x - 3 y = 7

(C)

3 x + 2 y = 5

(D)

2 x + 3 y = 9

JEE Main 2003

LEVELJEE Main

Locus of centroid of the triangle whose vertices are $(a cos t, a sin t), (b sin t, - b cos t)$ and $(1, 0)$ , where $t$ is a parameter, is

(A)

(3 x + 1)^{2} + (3 y)^{2} = a^{2} - b^{2}

(B)

(3 x - 1)^{2} + (3 y)^{2} = a^{2} - b^{2}

(C)

(3 x - 1)^{2} + (3 y)^{2} = a^{2} + b^{2}

(D)

(3 x + 1)^{2} + (3 y)^{2} = a^{2} + b^{2}

JEE Advanced 2009

LEVELJEE Main

The locus of the orthocentre of the triangle formed by the lines $(1 + p) x - p y + p (1 + p) = 0$ , $(1 + q) x - q y + q (1 + q) = 0$ , and $y = 0$ , where $p \neq = q$ , is

Let $O (0, 0), P (3, 4), Q (6, 0)$ be the vertices of the triangles $O P Q$ . The point $R$ inside the triangle $O P Q$ is such that the triangles $O P R, P QR, O QR$ are of equal area. The coordinates of $R$ are

(A)

(4/3, 3)

(B)

(3, 2/3)

(C)

(3, 4/3)

(D)

(4/3, 2/3)

JEE Advanced 2003

LEVELJEE Main

Orthocentre of triangle with vertices $(0, 0), (3, 4)$ and $(4, 0)$ is

(A)

(3, 5/4)

(B)

(3, 12)

(C)

(3, 3/4)

(D)

(3, 9)

JEE Advanced 2000

LEVELBoard

The incentre of the triangle with vertices $(1, 3), (0, 0)$ and $(2, 0)$ is

(A)

(1, \frac{3}{2})

(B)

(\frac{2}{3}, \frac{1}{3})

(C)

(\frac{2}{3}, \frac{3}{2})

(D)

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

JEE Advanced 1998

LEVELJEE Main

If the vertices $P, Q, R$ of a triangle $P QR$ are rational points, which of the following points of the triangle $P QR$ is (are) always rational point(s)?

* Multiple Correct Options

The orthocentre of the triangle formed by the lines $x y = 0$ and $x + y = 1$ is

(A)

(1/2, 1/2)

(B)

(1/3, 1/3)

(C)

(0, 0)

(D)

(1/4, 1/4)

JEE Advanced 1979

LEVELJEE Main

If the bisector of the angle $P$ of a triangle $P QR$ meets $QR$ in $S$ , then

(A)

QS = S R

(B)

QS : S R = P R : P Q

(C)

QS : S R = P Q : P R

The orthocentre of the triangle formed by the lines $x + y = 1, 2 x + 3 y = 6$ and $4 x - y + 4 = 0$ lies in quadrant number .........

JEE Advanced 1983

LEVELJEE Main

The vertices of a triangle are $[a t_{1} t_{2}, a (t_{1} + t_{2})]$ , $[a t_{2} t_{3}, a (t_{2} + t_{3})]$ , $[a t_{3} t_{1}, a (t_{3} + t_{1})]$ . Find the orthocentre of the triangle.

JEE Advanced 1979

LEVELJEE Advanced

(a) Two vertices of a triangle are $(5, - 1)$ and $(- 2, 3)$ . If the orthocentre of the triangle is the origin, find the coordinates of the third point. (b) Find the equation of the line which bisects the obtuse angle between the lines $x - 2 y + 4 = 0$ and $4 x - 3 y + 2 = 0$ .

JEE Advanced 1998

LEVELJEE Main

Using co-ordinate geometry, prove that the three altitudes of any triangle are concurrent.

JEE Main 2017

LEVELJEE Main

Let k be an integer such that the triangle with vertices $(k, - 3 k)$ , $(5, k)$ and $(- k, 2)$ has area 28 sq. units. Then the orthocentre of this triangle is at the point:

(A)

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

(B)

(1, \frac{3}{4})

(C)

(1, - \frac{3}{4})

(D)

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

JEE Main 2018 (Paper 1)

LEVELJEE Main

Let the orthocentre and centroid of a triangle be $A (- 3, 5)$ and $B (3, 3)$ respectively. If $C$ is the circumcentre of this triangle, then the radius of the circle having line segment $A C$ as diameter, is :

(A)

\frac{3 5}{2}

(B)

10

JEE Main 2018 (15 April Shift 1)

LEVELJEE Main

In a triangle ABC, coordinates of A are (1, 2) and the equations of the medians through B and C are respectively, $x + y = 5$ and $x = 4$ . Then area of $Δ A B C$ (in sq. units) is:

JEE Main 2019 (10 January Shift 1)

LEVELJEE Main

If the line $3 x + 4 y - 24 = 0$ intersects the x-axis at the point A and the y-axis at the point B, then the incentre of the triangle OAB, where O is the origin, is

JEE Main 2019 (10 January Shift 1)

LEVELJEE Main

A point P moves on the line $2 x - 3 y + 4 = 0$ . If Q(1, 4) and R(3, -2) are fixed points, then the locus of the centroid of $Δ P QR$ is a line :

(A)

parallel to x-axis

(B)

with slope \frac{2}{3}

(C)

with slope \frac{3}{2}

(D)

parallel to y-axis

JEE Main 2019 (9 January)

LEVELJEE Advanced

Let the equations of two sides of a triangle be $3 x - 2 y + 6 = 0$ and $4 x + 5 y - 20 = 0$ . If the orthocentre of this triangle is at (1,1), then the equation of its third side is :

(A)

122 y - 26 x - 1675 = 0

(B)

26 x + 61 y + 1675 = 0

(C)

122 y + 26 x + 1675 = 0

(D)

26 x - 122 y - 1675 = 0

JEE Main 2019 (10 January)

LEVELJEE Main

Two vertices of a triangle are $(0, 2)$ and $(4, 3)$ . If its orthocentre is at the origin, then its third vertex lies in which quadrant ?

JEE Main 2019 (12 April)

LEVELJEE Main

A triangle has a vertex at (1, 2) and the mid points of the two sides through it are (-1, 1) and (2, 3). Then the centroid of this triangle is :

JEE Main 2020 (7 January Shift 1)

LEVELJEE Main

Let $A (1, 0), B (6, 2), C (\frac{3}{2}, 6)$ be the vertices of a triangle $A B C$ . If $P$ is a point inside the triangle $A B C$ such that the triangles $A P C, A P B$ and $B P C$ have equal areas, then the length of the line segment $P Q$ , where $Q$ is the point $(- \frac{7}{6}, - \frac{1}{3})$ , is

JEE Main 2020 - 7 Jan (Morning)

LEVELJEE Main

Let $A (1, 0)$ , $B (6, 2)$ and $C (\frac{3}{2}, 6)$ be the vertices of a triangle $A B C$ . If $P$ is a point inside the triangle $A B C$ such that the triangles $A P C$ , $A P B$ and $B P C$ have equal areas, then the length of the line segment $P Q$ , where $Q$ is the point $(- \frac{7}{6}, - \frac{1}{3})$ , is.....

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The x-coordinate of the incentre of the triangle that has the coordinates of mid points of its sides as (0,1),(1,1) and (1,0) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a vertex of a triangle is (1,1) and the mid points of two sides through this vertex are (−1,2) and (3,2) then the centroid of the triangle is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let A(2,−3) and B(−2,3) be vertices of a triangle ABC. If the centroid of this triangle moves on the line 2x+3y=1, then the locus of the vertex C is the line

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Locus of centroid of the triangle whose vertices are (acost,asint),(bsint,−bcost) and (1,0), where t is a parameter, is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The locus of the orthocentre of the triangle formed by the lines (1+p)x−py+p(1+p)=0, (1+q)x−qy+q(1+q)=0, and y=0, where p=q, is

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

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The incentre of the triangle with vertices (1,3​),(0,0) and (2,0) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the vertices P,Q,R of a triangle PQR are rational points, which of the following points of the triangle PQR is (are) always rational point(s)?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The orthocentre of the triangle formed by the lines xy=0 and x+y=1 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the bisector of the angle P of a triangle PQR meets QR in S, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The orthocentre of the triangle formed by the lines x+y=1,2x+3y=6 and 4x−y+4=0 lies in quadrant number .........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The vertices of a triangle are [at1​t2​,a(t1​+t2​)], [at2​t3​,a(t2​+t3​)], [at3​t1​,a(t3​+t1​)]. Find the orthocentre of the triangle.

Using co-ordinate geometry, prove that the three altitudes of any triangle are concurrent.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let k be an integer such that the triangle with vertices (k,−3k), (5,k) and (−k,2) has area 28 sq. units. Then the orthocentre of this triangle is at the point:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } In a triangle ABC, coordinates of A are (1, 2) and the equations of the medians through B and C are respectively, x+y=5 and x=4. Then area of ΔABC (in sq. units) is:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the line 3x+4y−24=0 intersects the x-axis at the point A and the y-axis at the point B, then the incentre of the triangle OAB, where O is the origin, is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A point P moves on the line 2x−3y+4=0. If Q(1, 4) and R(3, -2) are fixed points, then the locus of the centroid of ΔPQR is a line :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let the equations of two sides of a triangle be 3x−2y+6=0 and 4x+5y−20=0. If the orthocentre of this triangle is at (1,1), then the equation of its third side is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Two vertices of a triangle are (0,2) and (4,3). If its orthocentre is at the origin, then its third vertex lies in which quadrant ?

A triangle has a vertex at (1, 2) and the mid points of the two sides through it are (-1, 1) and (2, 3). Then the centroid of this triangle is :

The x-coordinate of the incentre of the triangle that has the coordinates of mid points of its sides as $(0, 1), (1, 1)$ and $(1, 0)$ is

If a vertex of a triangle is $(1, 1)$ and the mid points of two sides through this vertex are $(- 1, 2)$ and $(3, 2)$ then the centroid of the triangle is

Let $A (2, - 3)$ and $B (- 2, 3)$ be vertices of a triangle $A B C$ . If the centroid of this triangle moves on the line $2 x + 3 y = 1$ , then the locus of the vertex $C$ is the line

Locus of centroid of the triangle whose vertices are $(a cos t, a sin t), (b sin t, - b cos t)$ and $(1, 0)$ , where $t$ is a parameter, is

The locus of the orthocentre of the triangle formed by the lines $(1 + p) x - p y + p (1 + p) = 0$ , $(1 + q) x - q y + q (1 + q) = 0$ , and $y = 0$ , where $p \neq = q$ , is

Orthocentre of triangle with vertices $(0, 0), (3, 4)$ and $(4, 0)$ is

The incentre of the triangle with vertices $(1, 3), (0, 0)$ and $(2, 0)$ is

If the vertices $P, Q, R$ of a triangle $P QR$ are rational points, which of the following points of the triangle $P QR$ is (are) always rational point(s)?

The orthocentre of the triangle formed by the lines $x y = 0$ and $x + y = 1$ is

If the bisector of the angle $P$ of a triangle $P QR$ meets $QR$ in $S$ , then

The orthocentre of the triangle formed by the lines $x + y = 1, 2 x + 3 y = 6$ and $4 x - y + 4 = 0$ lies in quadrant number .........

The vertices of a triangle are $[a t_{1} t_{2}, a (t_{1} + t_{2})]$ , $[a t_{2} t_{3}, a (t_{2} + t_{3})]$ , $[a t_{3} t_{1}, a (t_{3} + t_{1})]$ . Find the orthocentre of the triangle.

Let k be an integer such that the triangle with vertices $(k, - 3 k)$ , $(5, k)$ and $(- k, 2)$ has area 28 sq. units. Then the orthocentre of this triangle is at the point:

In a triangle ABC, coordinates of A are (1, 2) and the equations of the medians through B and C are respectively, $x + y = 5$ and $x = 4$ . Then area of $Δ A B C$ (in sq. units) is:

If the line $3 x + 4 y - 24 = 0$ intersects the x-axis at the point A and the y-axis at the point B, then the incentre of the triangle OAB, where O is the origin, is

A point P moves on the line $2 x - 3 y + 4 = 0$ . If Q(1, 4) and R(3, -2) are fixed points, then the locus of the centroid of $Δ P QR$ is a line :

Let the equations of two sides of a triangle be $3 x - 2 y + 6 = 0$ and $4 x + 5 y - 20 = 0$ . If the orthocentre of this triangle is at (1,1), then the equation of its third side is :

Two vertices of a triangle are $(0, 2)$ and $(4, 3)$ . If its orthocentre is at the origin, then its third vertex lies in which quadrant ?