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Consider the triangle $∆ O A B$ whose sides $O A$ and $O B$ are represented by $y^{2} = m^{2} x^{2} .$ If $H (α, β)$ is the orthocentre, then equation of $A B$ is given by_____

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Important Questions on Point and Straight Line

MEDIUM

Mathematics>Coordinate Geometry>Point and Straight Line>Special Points in Triangles

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} =

HARD

Mathematics>Coordinate Geometry>Point and Straight Line>Special Points in Triangles

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:

HARD

Mathematics>Coordinate Geometry>Point and Straight Line>Special Points in Triangles

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>Coordinate Geometry>Point and Straight Line>Special Points in Triangles

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

EASY

Mathematics>Coordinate Geometry>Point and Straight Line>Special Points in Triangles

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:

EASY

Mathematics>Coordinate Geometry>Point and Straight Line>Special Points in Triangles

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

MEDIUM

Mathematics>Coordinate Geometry>Point and Straight Line>Special Points in Triangles

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>Coordinate Geometry>Point and Straight Line>Special Points in Triangles

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)

HARD

Mathematics>Coordinate Geometry>Point and Straight Line>Special Points in Triangles

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>Coordinate Geometry>Point and Straight Line>Special Points in Triangles

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}),

HARD

Mathematics>Coordinate Geometry>Point and Straight Line>Special Points in Triangles

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:

HARD

Mathematics>Coordinate Geometry>Point and Straight Line>Special Points in Triangles

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>Coordinate Geometry>Point and Straight Line>Special Points in Triangles

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 :

MEDIUM

Mathematics>Coordinate Geometry>Point and Straight Line>Special Points in Triangles

The circumcentre of the triangle with vertices at

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

and

(2, 1)

HARD

Mathematics>Coordinate Geometry>Point and Straight Line>Special Points in Triangles

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>Coordinate Geometry>Point and Straight Line>Special Points in Triangles

If the sides of a triangle are

3, 4

and

5

then the circumradius of the triangle is

HARD

Mathematics>Coordinate Geometry>Point and Straight Line>Special Points in Triangles

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>Coordinate Geometry>Point and Straight Line>Special Points in Triangles

R

is the circum radius of

Δ A B C

, then

A r e a (Δ A B C)

= ….

EASY

Mathematics>Coordinate Geometry>Point and Straight Line>Special Points in Triangles

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

HARD

Mathematics>Coordinate Geometry>Point and Straight Line>Special Points in Triangles

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:

Consider the triangle ∆OAB whose sides OA and OB are represented by y2=m2x2. If Hα, β is the orthocentre, then equation of AB is given by_____

Important Questions on Point and Straight Line

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Consider the triangle $∆ O A B$ whose sides $O A$ and $O B$ are represented by $y^{2} = m^{2} x^{2} .$ If $H (α, β)$ is the orthocentre, then equation of $A B$ is given by_____