Two side of rhombus lying in the 1 st quadrant are given by y = 3 x 4 and y = 4 x 3 . If the length of the longer diagonal O C = 12 units, find the equation of other two sides.

We have, Here, A ≡ 0 , 0 A C is the longer diagonal. A C is one of the bisectors of A B and A D , so 3 x - 4 y 5 = ± 3 y - 4 x 5 ⇒ 3 x - 4 y = ± 3 y - 4 x So, y = x and y = - x . Since, rhombus is in the first quadrant, so y = x is the equation of A C . So, let C ≡ r , r , hence A C = 12 ⇒ r 2 + r 2 = 12 ⇒ 2 r 2 = 144 ⇒ r 2 = 72 ⇒ r = 6 2 Since, r > 0 , so C ≡ 6 2 , 6 2 . Now, Now write the equation of the lines through C which are parallel to given lines whose slopes are 3 4 and 4 3 by y - 6 2 = 3 4 x - 6 2 & y - 6 2 = 4 3 x - 6 2 ⇒ 3 x - 4 y + 6 2 & 4 x - 3 y - 6 2 = 0

Two side of rhombus lying in the 1 st quadrant are given by y = 3 x 4 and y = 4 x 3 . If the length of the longer diagonal O C = 12 units, find the equation of other two sides.

We have, Here, A ≡ 0 , 0 A C is the longer diagonal. A C is one of the bisectors of A B and A D , so 3 x - 4 y 5 = ± 3 y - 4 x 5 ⇒ 3 x - 4 y = ± 3 y - 4 x So, y = x and y = - x . Since, rhombus is in the first quadrant, so y = x is the equation of A C . So, let C ≡ r , r , hence A C = 12 ⇒ r 2 + r 2 = 12 ⇒ 2 r 2 = 144 ⇒ r 2 = 72 ⇒ r = 6 2 Since, r > 0 , so C ≡ 6 2 , 6 2 . Now, Now write the equation of the lines through C which are parallel to given lines whose slopes are 3 4 and 4 3 by y - 6 2 = 3 4 x - 6 2 & y - 6 2 = 4 3 x - 6 2 ⇒ 3 x - 4 y + 6 2 & 4 x - 3 y - 6 2 = 0

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Two side of rhombus lying in the $1^{st}$ quadrant are given by $y = \frac{3 x}{4}$ and $y = \frac{4 x}{3}$ . If the length of the longer diagonal $O C = 12$ units, find the equation of other two sides.

Important Questions on Point and Straight Line

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Mathematics>Coordinate Geometry>Point and Straight Line>Equation of the Bisector of the Angle Between Two Lines

Let $A$ be a fixed point $\Rightarrow 3 x - 4 y + 6 \sqrt{2} & 4 x - 3 y - 6 \sqrt{2} = 0$ and $B$ be a moving point $(2 t, 0) .$ Let $M$ be the mid-point of $A B$ and the perpendicular bisector of $A B$ meets the $y -$ axis at $C .$ The locus of the mid-point $P$ of $MC$ is

EASY

Mathematics>Coordinate Geometry>Point and Straight Line>Equation of the Bisector of the Angle Between Two Lines

Consider a

Δ P Q R

in which the relation

Q R^{2} + P R^{2} = 5 P Q^{2}

holds. Let

G

be the point of intersection of medians

P M

and

Q N

. Then,

∠ Q G M

is always

HARD

Mathematics>Coordinate Geometry>Point and Straight Line>Equation of the Bisector of the Angle Between Two Lines

A ray travelling along the line

3 x + 4 y = 11

, after being reflected from a line

l

travels along the line

12 x - 5 y = 2

. Then one of the equations of the line

l

EASY

Mathematics>Coordinate Geometry>Point and Straight Line>Equation of the Bisector of the Angle Between Two Lines

In $△ P Q R$ , the bisector of $∠ P$ intersects $Q R$ in $M$ . If $P Q = 10, P R = 12, Q M = 8$ then $Q R =$ _____.

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Mathematics>Coordinate Geometry>Point and Straight Line>Equation of the Bisector of the Angle Between Two Lines

If the line

y = m x

is one of the bisectors of

x^{2} + 4 x y - y^{2} = 0,

then value of

2 m =

MEDIUM

Mathematics>Coordinate Geometry>Point and Straight Line>Equation of the Bisector of the Angle Between Two Lines

∆ A B C

\vec{A D}

is the bisector of

∠ A

intersects

\bar{B C}

D

∴ B D =

_____

MEDIUM

Mathematics>Coordinate Geometry>Point and Straight Line>Equation of the Bisector of the Angle Between Two Lines

In a triangle

A B C, ∠ B A C = 90 °; A D

is the altitude from

A

on to

B C

. Draw

D E

perpendicular to

A C

and

D F

perpendicular to

A B

. Suppose

A B = 15

and

B C = 25 .

Then the length of

E F

EASY

Mathematics>Coordinate Geometry>Point and Straight Line>Equation of the Bisector of the Angle Between Two Lines

In the given figure, $A D$ is the bisector of $∠ B A C$ . If $A B = 10 c m, A C = 6 c m$ and $B C = 12 c m$ . Find $B D$ .

Question Image

MEDIUM

Mathematics>Coordinate Geometry>Point and Straight Line>Equation of the Bisector of the Angle Between Two Lines

If the straight line

2 x + 3 y + 1 = 0

bisects the angle between a pair of lines, one of which in this pair is

3 x + 2 y + 4 = 0

. Then, the equation of the other line in that pair of lines is

EASY

Mathematics>Coordinate Geometry>Point and Straight Line>Equation of the Bisector of the Angle Between Two Lines

Let $a, b, c$ be the side-length of a triangle and $l, m, n$ be the lengths of its medians. Put $K = \frac{l + m + n}{a + b + c}$ Then, as $a, b, c$ vary, $K$ can assume every value in the interval

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Mathematics>Coordinate Geometry>Point and Straight Line>Equation of the Bisector of the Angle Between Two Lines

D

and

E

are the points on sides

B C

and

A C

, respectively of

∆ A B C . A D

and

B E

intersect each other at T. If

\frac{A T}{T D} = 5

and

\frac{B T}{E T} = 7

, then

C D : B D =

HARD

Mathematics>Coordinate Geometry>Point and Straight Line>Equation of the Bisector of the Angle Between Two Lines

Two equal sides of an isosceles triangle are

7 x - y + 3 = 0

and

x + y - 3 = 0

and its third side passes through the point

(1, - 10)

. The equation of the third side is

EASY

Mathematics>Coordinate Geometry>Point and Straight Line>Equation of the Bisector of the Angle Between Two Lines

∆ ABC

∠ ABC = 90 °

and

∠ BAC = 60 °

. If bisector of

∠ BAC

meets

BC

D

. Then

BD : DC

HARD

Mathematics>Coordinate Geometry>Point and Straight Line>Equation of the Bisector of the Angle Between Two Lines

Question Image

In $△ A B C$ , it is given that $\frac{B D}{D C} = \frac{A B}{A C}, ∠ B = 70 °, ∠ C = 50 ° .$ Find $∠ B A D$ in degrees.

MEDIUM

Mathematics>Coordinate Geometry>Point and Straight Line>Equation of the Bisector of the Angle Between Two Lines

In the given fig. $A M ⊥ B C$ and $A N$ is the bisector of $∠ A$ . If $∠ B = 65 °$ and $∠ C = 33 °$ , then the value of $∠ M A N$ will be

Question Image

HARD

Mathematics>Coordinate Geometry>Point and Straight Line>Equation of the Bisector of the Angle Between Two Lines

Let

m_{1}, m_{2}

be the slopes of two adjacent sides of a square of side

a

such that

a^{2} + 11 a + 3 (m_{1}^{2} + m_{2}^{2}) = 220

. If one vertex of the square is

(10 (\cos α - \sin α), 10 (\sin α + \cos α))

, where

α \in (0, \frac{π}{2})

and the equation of one diagonal is

(cosα - sinα) x + (\sin α + \cos α) y = 10

, then

72 (\sin^{4} α + \cos^{4} α) + a^{2} - 3 a + 13

is equal to

MEDIUM

Mathematics>Coordinate Geometry>Point and Straight Line>Equation of the Bisector of the Angle Between Two Lines

Let the point

P (α, β)

be at a unit distance from each of the two lines

L_{1} : 3 x - 4 y + 12 = 0

, and

L_{2} : 8 x + 6 y + 11 = 0

. If

P

lies below

L_{1}

and above

L_{2}

, then

100 (α + β)

is equal to

MEDIUM

Mathematics>Coordinate Geometry>Point and Straight Line>Equation of the Bisector of the Angle Between Two Lines

Let

P Q R

be an acute-angled triangle in which

P Q < Q R

. From the vertex

Q

draw the altitude

Q Q_{1}

the angle bisector

Q Q_{2}

and the median

Q Q_{3}

, with

Q_{1,} Q_{2,} Q_{3}

lying on

P R

. Then,

MEDIUM

Mathematics>Coordinate Geometry>Point and Straight Line>Equation of the Bisector of the Angle Between Two Lines

Statement-I: Two lines which pass through a given fixed point and are equally inclined to two other lines passing through the same point, are always perpendicular to each other.
Statement-II: Angle bisectors of two intersecting lines are always perpendicular to each other.

MEDIUM

Mathematics>Coordinate Geometry>Point and Straight Line>Equation of the Bisector of the Angle Between Two Lines

The equation of the plane which bisects the angle between the planes

3 x - 6 y + 2 z + 5 = 0

and

4 x - 12 y + 3 z - 3 = 0

which contains the origin is

Two side of rhombus lying in the 1st quadrant are given by y=3x4 and y=4x3. If the length of the longer diagonal OC=12 units, find the equation of other two sides.

Important Questions on Point and Straight Line

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Two side of rhombus lying in the $1^{st}$ quadrant are given by $y = \frac{3 x}{4}$ and $y = \frac{4 x}{3}$ . If the length of the longer diagonal $O C = 12$ units, find the equation of other two sides.