EASY

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The coordinates of the orthocentre of an equilateral triangle formed by the vertices $A (a_{1}, b_{2}), B (a_{2}, b_{2})$ and $C (a_{3}, b_{3})$ is

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Important Questions on Coordinate Geometry

EASY

Mathematics>Geometry>Coordinate Geometry>Special Points in a Triangle

P (0, 0), Q (1, 0)

and

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

are three given points, then the centre of the circle for which the lines

P Q, Q R

and

R P

are the tangents is

HARD

Mathematics>Geometry>Coordinate Geometry>Special Points in a Triangle

The distance (in units) between the circumcentre and the centroid of the triangle formed by the vertices $(1, 2), (3, - 1)$ and $(4, 0),$ is

MEDIUM

Mathematics>Geometry>Coordinate Geometry>Special Points in a Triangle

The circumcentre of the triangle with vertices at

(- 2, 3), (1, - 2)

and

(2, 1)

HARD

Mathematics>Geometry>Coordinate Geometry>Special Points in a Triangle

Let

\tan α, \tan β

and

\tan γ; α, β, γ \neq \frac{(2 n - 1) π}{2}, n \in N

be the slopes of the three line segments

O A, O B

and

O C,

respectively, where

O

is origin. If circumcentre of

Δ A B C

coincides with origin and its orthocentre lies on

y

-axis, then the value of

{(\frac{\cos 3 α + \cos 3 β + \cos 3 γ}{\cos α \cdot \cos β \cdot \cos γ})}^{2}

is equal to :

EASY

Mathematics>Geometry>Coordinate Geometry>Special Points in a Triangle

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:

HARD

Mathematics>Geometry>Coordinate Geometry>Special Points in a Triangle

Let

a, b, c

be in arithmetic progression. Let the centroid of the triangle with vertices

(a, c), (2, b)

and

(a, b)

(\frac{10}{3}, \frac{7}{3}) .

α, β

are the roots of the equation

a x^{2} + b x + 1 = 0,

then the value of

α^{2} + β^{2} - α β

is:

MEDIUM

Mathematics>Geometry>Coordinate Geometry>Special Points in a Triangle

Let a triangle

A B C

be inscribed in a circle of radius

2

units. If the

3

bisectors of the angles

A, B

and

C

are extended to cut the circle at

A_{1}, B_{1}

and

C_{1}

respectively, then the value of

{[\frac{A A_{1} \cos \frac{A}{2} + B B_{1} \cos \frac{B}{2} + C C_{1} \cos \frac{C}{2}}{\sin A + \sin B + \sin C}]}^{2} =

EASY

Mathematics>Geometry>Coordinate Geometry>Special Points in a Triangle

If the sides of a triangle are

3, 4

and

5

then the circumradius of the triangle is

HARD

Mathematics>Geometry>Coordinate Geometry>Special Points in a Triangle

Let

O

be the origin and let

P Q R

be an arbitrary triangle. The point

S

is such that

\vec{O P} . \vec{O Q} + \vec{O R} . \vec{O S} = \vec{O R} . \vec{O P} + \vec{O Q} . \vec{O S} = \vec{O Q} . \vec{O R} + \vec{O P} . \vec{O S}

then triangle

P Q R

has

S

as its

EASY

Mathematics>Geometry>Coordinate Geometry>Special Points in a Triangle

R

is the circum radius of

Δ A B C

, then

A r e a (Δ A B C)

= ….

EASY

Mathematics>Geometry>Coordinate Geometry>Special Points in a Triangle

Let the centroid of an equilateral triangle

A B C

be at the origin. Let one of the sides of the equilateral triangle be along the straight line

x + y = 3

. If

R

and

r

be the radius of circumcircle and incircle respectively of

Δ A B C

, then

(R + r)

is equal to :

HARD

Mathematics>Geometry>Coordinate Geometry>Special Points in a 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 orthocenter of this triangle is at the point:

MEDIUM

Mathematics>Geometry>Coordinate Geometry>Special Points in a Triangle

Let

D, E, F

be points on the sides

B C, C A, A B

of a triangle

A B C,

respectively. Suppose

A D, B E, C F

are concurrent at

P .

\frac{P F}{P C} = \frac{2}{3}, \frac{P E}{P B} = \frac{2}{7}

and

\frac{P D}{P A} = \frac{m}{n}

where

m, n

are positive integers with

gcd (m, n) = 1 .

Find

(m + n)

MEDIUM

Mathematics>Geometry>Coordinate Geometry>Special Points in a Triangle

The point

Q

is the image of the point

P (1, 5)

about the line

y = x

and

R

is the image of the point

Q

about the line

y = - x

. The circumcentre of the

Δ P Q R

HARD

Mathematics>Geometry>Coordinate Geometry>Special Points in a Triangle

Let the straight lines

5 x - 3 y + 15 = 0

and

5 x + 3 y - 15 = 0

form a triangle with the

X

-axis. Then, the radius of the circle circumscribing this triangle is

EASY

Mathematics>Geometry>Coordinate Geometry>Special Points in a Triangle

The orthocentre of the triangle formed by the lines

x = 2, y = 3

and

3 x + 2 y = 6

is at the point

EASY

Mathematics>Geometry>Coordinate Geometry>Special Points in a Triangle

The vertices of a triangle are

(3, - 5)

and

(- 7, 4)

. If its centroid is

(2, - 1),

then the third vertex is

MEDIUM

Mathematics>Geometry>Coordinate Geometry>Special Points in a Triangle

Let

D

be the centroid of the triangle with vertices

(3, - 1)

(1, 3)

and

(2, 4)

. Let

P

be the point of intersection of the lines

x + 3 y - 1 = 10

and

3 x - y + 1 = 0

. Then, the line passing through the points

D

and

P

also passes through the point:

HARD

Mathematics>Geometry>Coordinate Geometry>Special Points in a Triangle

Let

P

be a point inside a triangle

A B C

with

∠ A B C = 90 °

. Let

P_{1} and P_{2}

be the images of

P

under reflection in

A B

and

B C

respectively. The distance between the circumcentre of triangles

A B C

and

P_{1} P P_{2}

HARD

Mathematics>Geometry>Coordinate Geometry>Special Points in a Triangle

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 orthocenter of this triangle is at

(1, 1)

then the equation of it's third side is:

The coordinates of the orthocentre of an equilateral triangle formed by the vertices Aa1,b2, Ba2,b2 and Ca3,b3 is

Important Questions on Coordinate Geometry

Learn and Prepare for any exam you want !

The coordinates of the orthocentre of an equilateral triangle formed by the vertices $A (a_{1}, b_{2}), B (a_{2}, b_{2})$ and $C (a_{3}, b_{3})$ is