Prove that the coordinates of the center of the circle inscribed in the triangle whose angular points are 1 , 2 , 2 , 3 and 3 , 1 are 8 + 10 6 , 16 - 10 6 . Also, find the coordinates of the centers of the escribed circles.

Let ABC be a triangle such that coordinates of A , B , and C are 1 , 2 , 2 , 3 and 3 , 1 respectively Using the distance formula length between two points x 1 , y 1 and x 2 , y 2 can be given as x 1 - x 2 2 + y 1 - y 2 2 So, length of BC = a = 3 - 2 2 + 1 - 3 2 = 5 Length of CA = b = 3 - 1 2 + 1 - 2 2 = 5 Length of AB = c = 2 - 1 2 + 3 - 2 2 = 2 In a triangle ABC , A x 1 , y 1 , B x 2 , y 2 and C x 3 , y 3 and sides BC = a , CA = b and AB = c then Coordinates of incenter are a x 1 + b x 2 + c x 3 a + b + c , a y 1 + b y 2 + c y 3 a + b + c and coordinates of excenters are - a x 1 + b x 2 + c x 3 - a + b + c , - a y 1 + b y 2 + c y 3 - a + b + c , a x 1 - b x 2 + c x 3 a - b + c , a y 1 - b y 2 + c y 3 a - b + c and a x 1 + b x 2 - c x 3 a + b - c , a y 1 + b y 2 - c y 3 a + b - c Therefore, coordinates of incenter are 5 × 1 + 5 × 2 + 2 × 3 5 + 5 + 2 , 5 × 2 + 5 × 3 + 2 × 1 5 + 5 + 2 = 3 5 + 3 2 2 5 + 2 , 5 5 + 2 2 5 + 2 = 3 5 + 3 2 2 5 - 2 2 5 + 2 2 5 - 2 , 5 5 + 2 2 5 - 2 2 5 + 2 2 5 - 2 8 + 10 6 , 6 - 10 6 (on simplifying) and coordinates of the excenter opposite to vertex A are ≡ - 5 + 2 5 + 3 2 - 5 + 5 + 2 , - 2 5 + 3 5 + 2 - 5 + 5 + 2 = 6 + 10 2 , 2 + 10 2 Coordinates of the excenter opposite to vertex B are ≡ 5 - 2 5 + 3 2 5 - 5 + 2 , 2 5 - 3 5 + 2 5 - 5 + 2 = 6 - 10 2 , 2 - 10 2 and coordinates of the excenter opposite to vertex C are 5 + 2 5 - 3 2 5 + 5 - 2 , 2 5 + 3 5 - 2 5 + 5 - 2 = 8 - 10 6 , 16 + 10 6

HARD

JEE Advanced

IMPORTANT

Earn 100

Through a given point $O$ , a straight line is drawn to meet two given parallel straight lines in $P$ and $Q$ ; through $P$ and $Q$ straight lines are drawn in a given direction to meet in $R$ . Prove that the locus of $R$ is a straight line.

Important Questions on The Straight Line (Continued)

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 4 - The Straight Line (Continued)>EXAMPLES X>Q 1

A moving straight line is such that the algebraic sum of the perpendiculars upon it from some of fixed points is zero; show that it always passes through a fixed point.

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 4 - The Straight Line (Continued)>EXAMPLES X>Q 2

Two fixed straight lines

O X

and

O Y

are cut by a variable line at the points

A

and

B

respectively, and

P

and

Q

are the feet of the perpendiculars drawn from

A

and

B

upon the lines

O B Y

and

O A X

respectively. Show that, if

A B

passes through a fixed point, then

P Q

will also pass through a fixed point.

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 4 - The Straight Line (Continued)>EXAMPLES X>Q 3

If the equal sides

A B

and

A C

of an isosceles triangle be produced to

E

and

F

so that

B E . C F = A B^{2}

. Show that the line

E F

will always pass through a fixed point.

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 4 - The Straight Line (Continued)>EXAMPLES X>Q 4

If a straight line

a x + b y + c = 0, a, b, c > 0

moves so that the sum of the perpendiculars fall on it from the two fixed points

(3,4)

and

(7,2)

is equal to three times the perpendicular on it from a third fixed point

(1,3)

, prove that there is another fixed point through which this line always passes and find its coordinates.

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 4 - The Straight Line (Continued)>EXAMPLES X>Q 5

Find the center and radius of the circle which is inscribed in the triangle formed by the straight lines whose equations are

3 x + 4 y + 2 = 0, 3 x - 4 y + 12 = 0

and

4 x - 3 y = 0

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 4 - The Straight Line (Continued)>EXAMPLES X>Q 6

Find the center and radius of the circle which is inscribed in the triangle formed by the straight lines whose equations are

2 x + 4 y + 3 = 0, 4 x + 3 y + 3 = 0

and

x + 1 = 0

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 4 - The Straight Line (Continued)>EXAMPLES X>Q 7

Find the center and radius of the circle which is inscribed in the triangle formed by the straight lines whose equations are

y = 0, 12 x - 5 y = 0

and

3 x + 4 y - 7 = 0

MEDIUM

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 4 - The Straight Line (Continued)>EXAMPLES X>Q 8

Prove that the coordinates of the center of the circle inscribed in the triangle whose angular points are

(1, 2), (2, 3)

and

(3, 1)

are

(\frac{(8 + \sqrt{10})}{6}, \frac{(16 - \sqrt{10})}{6})

. Also, find the coordinates of the centers of the escribed circles.

Through a given point O, a straight line is drawn to meet two given parallel straight lines in P and Q; through P and Q straight lines are drawn in a given direction to meet in R. Prove that the locus of R is a straight line.

Important Questions on The Straight Line (Continued)

Learn and Prepare for any exam you want !

Through a given point $O$ , a straight line is drawn to meet two given parallel straight lines in $P$ and $Q$ ; through $P$ and $Q$ straight lines are drawn in a given direction to meet in $R$ . Prove that the locus of $R$ is a straight line.