## Equations of Loci

 7.4 Equations of Loci

 Equation of Locus Locus of moving point $$P(x,y)$$ which distance is constant from a single fixed point $$A(x_1,y_1)$$ is: $$(x-x_1)^2+(y-y_1)^2=r^2$$ Locus of moving point $$P(x,y)$$ which distance from two fixed points $$A(x_1,y_1)$$ and $$B(x_2,y_2)$$ is always constant in the ratio $$m:n$$ is:  $$\dfrac{(x-x_1)^2+(y-y_1)^2}{(x-x_2)^2+(y-y_2)^2}=\dfrac{m^2}{n^2}$$ Locus of moving point $$P(x,y)$$ which distance from two fixed points $$A(x_1,y_1)$$ and $$B(x_2,y_2)$$ is the same is: $$(x-x_1)^2+(y-y_1)^2=(x-x_2)^2+(y-y_2)^2$$

Example $$1$$
 Question

Find the equation of the locus of a moving point $$P(x,y)$$ so that its distance is always $$5$$ units from a fixed point $$Q(2,4)$$.

 Solution

The formula involved is (a fixed point):

$$(x-x_1)^2+(y-y_1)^2=r^2$$

Substitute the value of the coordinate $$Q(2,4)$$ and the distance of $$5$$ units into the formula:

$$(x-2)^2+(y-4)^2=5^2$$

Expand and simplify the resulting equation:

$$x^2-4x+4+y^2-8y+16=25 \\ x^2+y^2-4x-8y-5=0$$

The equation of the locus of a moving point $$P$$ is:

$$x^2+y^2-4x-8y-5=0$$

Example $$2$$
 Question

Find the equation of the locus of the moving point $$P(x,y)$$ so that its distance from point $$A(-2,3)$$ and point $$B(4,-1)$$ is the same.

 Solution

The formula involved is (two fixed point):

$$(x-x_1)^2+(y-y_1)^2=(x-x_2)^2+(y-y_2)^2$$

Substitute the value of the coordinates $$A(-2,3)$$ and $$B(4,-1)$$ into the formula:

$$(x-(-2))^2+(y-3)^2=(x-4)^2+(y-(-1))^2$$

Expand and simplify the resulting equation:

$$(x+2)^2+(y-3)^2=(x-4)^2+(y+1)^2 \\ x^2+2x+4+y^2-6y+9=x^2-8x+16+y^2+2y+1 \\ 10x-8y-4=0$$

The equation of the locus of a moving point $$P$$ is:

$$10x-8y-4=0$$

Example $$3$$
 Question

$$A(2,0)$$ and $$B(0,-2)$$ are two fixed points. Point $$P$$ moves in a ratio such that $$AP:PB=1:2$$. Find the equation of the locus of the moving point $$P$$.

 Solution

The formula involved is (two fixed point, ratio):

$$\dfrac{(x-x_1)^2+(y-y_1)^2}{(x-x_2)^2+(y-y_2)^2}=\dfrac{m^2}{n^2}$$

Substitute the value of the coordinates $$A(2,0)$$ and $$B(4,-1)$$ with ratio $$AP:PB=1:2$$ into the formula:

$$\dfrac{AP}{PB}=\dfrac{1}{2}$$

$$\dfrac{(x-2)^2+(y-0)^2}{(x-0)^2+(y+2)^2}=\dfrac{1}{2^2}$$

Expand and simplify the resulting equation:

$$4[(x-2)^2+y^2]=x^2+(y+2)^2 \\ 4(x^2-4x+4)+4y^2=x^2+(y^2+4y+4) \\ 4x^2-16x+16+4y^2=x^2+y^2+4y+4 \\ 3x^2+3y^2-16x-4y+12=0$$

The equation of the locus of a moving point $$P$$ is:

$$3x^2+3y^2-16x-4y+12=0$$

## Equations of Loci

 7.4 Equations of Loci

 Equation of Locus Locus of moving point $$P(x,y)$$ which distance is constant from a single fixed point $$A(x_1,y_1)$$ is: $$(x-x_1)^2+(y-y_1)^2=r^2$$ Locus of moving point $$P(x,y)$$ which distance from two fixed points $$A(x_1,y_1)$$ and $$B(x_2,y_2)$$ is always constant in the ratio $$m:n$$ is:  $$\dfrac{(x-x_1)^2+(y-y_1)^2}{(x-x_2)^2+(y-y_2)^2}=\dfrac{m^2}{n^2}$$ Locus of moving point $$P(x,y)$$ which distance from two fixed points $$A(x_1,y_1)$$ and $$B(x_2,y_2)$$ is the same is: $$(x-x_1)^2+(y-y_1)^2=(x-x_2)^2+(y-y_2)^2$$

Example $$1$$
 Question

Find the equation of the locus of a moving point $$P(x,y)$$ so that its distance is always $$5$$ units from a fixed point $$Q(2,4)$$.

 Solution

The formula involved is (a fixed point):

$$(x-x_1)^2+(y-y_1)^2=r^2$$

Substitute the value of the coordinate $$Q(2,4)$$ and the distance of $$5$$ units into the formula:

$$(x-2)^2+(y-4)^2=5^2$$

Expand and simplify the resulting equation:

$$x^2-4x+4+y^2-8y+16=25 \\ x^2+y^2-4x-8y-5=0$$

The equation of the locus of a moving point $$P$$ is:

$$x^2+y^2-4x-8y-5=0$$

Example $$2$$
 Question

Find the equation of the locus of the moving point $$P(x,y)$$ so that its distance from point $$A(-2,3)$$ and point $$B(4,-1)$$ is the same.

 Solution

The formula involved is (two fixed point):

$$(x-x_1)^2+(y-y_1)^2=(x-x_2)^2+(y-y_2)^2$$

Substitute the value of the coordinates $$A(-2,3)$$ and $$B(4,-1)$$ into the formula:

$$(x-(-2))^2+(y-3)^2=(x-4)^2+(y-(-1))^2$$

Expand and simplify the resulting equation:

$$(x+2)^2+(y-3)^2=(x-4)^2+(y+1)^2 \\ x^2+2x+4+y^2-6y+9=x^2-8x+16+y^2+2y+1 \\ 10x-8y-4=0$$

The equation of the locus of a moving point $$P$$ is:

$$10x-8y-4=0$$

Example $$3$$
 Question

$$A(2,0)$$ and $$B(0,-2)$$ are two fixed points. Point $$P$$ moves in a ratio such that $$AP:PB=1:2$$. Find the equation of the locus of the moving point $$P$$.

 Solution

The formula involved is (two fixed point, ratio):

$$\dfrac{(x-x_1)^2+(y-y_1)^2}{(x-x_2)^2+(y-y_2)^2}=\dfrac{m^2}{n^2}$$

Substitute the value of the coordinates $$A(2,0)$$ and $$B(4,-1)$$ with ratio $$AP:PB=1:2$$ into the formula:

$$\dfrac{AP}{PB}=\dfrac{1}{2}$$

$$\dfrac{(x-2)^2+(y-0)^2}{(x-0)^2+(y+2)^2}=\dfrac{1}{2^2}$$

Expand and simplify the resulting equation:

$$4[(x-2)^2+y^2]=x^2+(y+2)^2 \\ 4(x^2-4x+4)+4y^2=x^2+(y^2+4y+4) \\ 4x^2-16x+16+4y^2=x^2+y^2+4y+4 \\ 3x^2+3y^2-16x-4y+12=0$$

The equation of the locus of a moving point $$P$$ is:

$$3x^2+3y^2-16x-4y+12=0$$