Question

two fixed points A (-2, 3) and B(5,3) are given. Find the equation of locus of a moving point P such that area of triangle PAB is 20 sq units

Answer 1

Given:

• Two fixed points A(-2, 3) and B(5,3)
• A movable point P such that area of ΔPAB is 20 sq units

To Find:

Equation of locus of given moving point P

Solution:

Let the coordinates of movable point P be (x,y)

We know that,

• For a triangle having coordinates of vertices as (x₁,y₁), (x₂,y₂) and (x₃,y₃)

$\sf{\pink{Area\:of\:\triangle=\Bigg|\dfrac{1}{2}\Big(x_{1}(y_{2}-y_{3})+x_{2}(y_{3}-y_{1})+x_{3}(y_{1}-y_{2})\Big)\Bigg|}}$

Now, in ΔPAB

Let

• P(x,y)⇒(x₁,y₁)

• A(-2,3)⇒(x₂,y₂)

• B(5,3)⇒(x₃,y₃)

So,

$\sf{Area\:of\:\triangle PAB=\dfrac{1}{2}\Bigg|\Big(x(3-3)+(-2)(3-y)+5(y-3)\Big)\Bigg|}$

$\sf{20=\dfrac{1}{2}\Bigg|\Big(x(0)-2(3-y)+5(y-3)\Big)\Bigg|}$

$\sf{\dfrac{1}{2}\Bigg|\Big(x(0)-2(3-y)+5(y-3)\Big)\Bigg|=20}$

$\sf{\Bigg|\Big(x(0)-2(3-y)+5(y-3)\Big)\Bigg|=40}$

$\sf{\Bigg|-6+2y+5y-15\Bigg|=40}$

$\sf{\Bigg|7y-21\Bigg|=40}$

$\sf{7y-21=\pm40}$

Case 1:

$\longrightarrow\sf{7y-21=40}$

$\longrightarrow\sf{7y=40+21}$

$\longrightarrow\sf{7y=61}$

$\longrightarrow\sf{y=\dfrac{61}{7}}$

Case 2:

$\longrightarrow\sf{7y-21=-40}$

$\longrightarrow\sf{7y=-40+21}$

$\longrightarrow\sf{7y=-19}$

$\longrightarrow\sf{y=\dfrac{-19}{7}}$

Hence, the locus of moving point P is y=61/7 and y= -19/7.

Answer 2

Given:

Two fixed points A(-2, 3) and B(5,3)

A movable point P such that area of ΔPAB is 20 sq units

To Find:

Equation of locus of given moving point P

Solution:

Let the coordinates of movable point P be (x,y)

We know that,

For a triangle having coordinates of vertices as (x₁,y₁), (x₂,y₂) and (x₃,y₃)

21 x 1 (y 2−y 3 )+x 2 (y 3−y 1 )+x 3 (y 1 −y 2))

Now, in ΔPAB

Let

P(x,y)⇒(x₁,y₁)

A(-2,3)⇒(x₂,y₂)

B(5,3)⇒(x₃,y₃)

So,

Areaof△PAB= 21

(x(3−3)+(−2)(3−y)+5(y−3))

21

(x(0)−2(3−y)+5(y−3))

=21

(x(0)−2(3−y)+5(y−3))

=20

(x(0)−2(3−y)+5(y−3))

=40

−6+2y+5y−15

=40

7y−21

=40

7y−21=±40

Case 1:

⟶7y−21=40

⟶7y=40+21

⟶7y=61

⟶y= 761

Case 2:

⟶7y−21=−40

⟶7y=−40+21

⟶7y=−19

⟶y= 7−19

Hence, the locus of moving point P is y=61/7 and y= -19/7.