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The orthocentre of a right-angled triangle lies

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Important Questions on Coordinate Geometry

HARD

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

If the orthocentre of the triangle, whose vertices are

(1, 2), (2, 3)

and

(3, 1)

(α, β)

, then the quadratic equation whose roots are

α + 4 β

and

4 α + β

, is

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 :

HARD

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

The equations of the sides

A B, B C

and

C A

of a triangle

A B C

are

2 x + y = 0, x + p y = 15 a

and

x - y = 3

respectively. If its orthocentre is

(2, a)

- \frac{1}{2} < a < 2

, then

p

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:

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

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}

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

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

HARD

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

If a

∆ A B C

has vertices

A (- 1, 7), B (- 7, 1)

and

C (5, - 5)

, then its orthocentre has coordinates:

MEDIUM

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

A line cuts the

x

-axis at

A (7, 0)

and the

y

-axis at

B (0, - 5)

. A variable line

P Q

is drawn perpendicular to

A B

cutting the

x

-axis at

P (a, 0)

and the

y

-axis at

Q (0, b)

. If

A Q

and

B P

intersect at

R

, the locus of

R

HARD

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

Let

A (1, 0), B (6, 2)

and

C (\frac{3}{2}, 6)

be the vertices of a triangle

A B C .

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}),

EASY

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

The circumcentre of a triangle lies at the origin and its centroid is the midpoint of the line segment joining the points

(a^{2} + 1, a^{2} + 1)

and

(2 a, - 2 a), a \neq0

. Then for any

a,

the orthocentre of this triangle lies on the line

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)

= ….

HARD

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

The orthocentre of the triangle having vertices

A (1, 2), B (3, - 4)

and

C (0, 6)

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:

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)

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

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 :

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 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 orthocentre of a right-angled triangle lies

Important Questions on Coordinate Geometry

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