Find the locus of a point p whose distances from two fixed points A, A' are in a constant ratio 1:M, that is |PA| : |PA'| = 1:M. ( M >0)

Trevor Rush
2022-08-08
Answered

You can still ask an expert for help

Jazmin Cameron

Answered 2022-08-09
Author has **16** answers

Let us suppose that P,A and A' has a coordinate (x,y),(x1,y2) and (x3,y3) respectively.

First find the distance between P and A and distance between Pand A' through distance formula.

DISTANCE B/W Pand A:

|PA|= sqrt((x1-x)^2+(y1-y)^2)

=sqrt((x^2-2x*x1+x1^2) +(y^2-2y*y1+y1^2))

|PA'|= sqrt((x2-x)^2+(y2-y)^2)

=sqrt((x^2-2x*x2+x2^2) +(y^2-2y*y2+y2^2))

Now Applying condition:

|PA| = 1

|PA'| M

sqrt((x^2-2x*x1+x1^2) +(y^2-2y*y1+y1^2)) = 1

sqrt((x^2-2x*x2+x2^2) +(y^2-2y*y2+y2^2)) M

Squaring on both sides

(x^2-2x*x1+x1^2) +(y^2-2y*y1+y1^2) = 1

(x^2-2x*x2+x2^2) +(y^2-2y*y2+y2^2) M

Cross Multiplying

M((x^2-2x*x1+x1^2) +(y^2-2y*y1+y1^2)) = (x^2-2x*x2+x2^2) +(y^2-2y*y2+y2^2)

M*x^2-2M*x*x1+M*x1^2+M*y^2-2*M*y*y1+M*y1^2 = (x^2-2x*x2+x2^2) +(y^2-2y*y2+y2^2)

M*x^2-x^2+M*y^2-y^2-2M*x*x1+2x*x2-2*M*y*y1+2y*y2+M*x1^2+M*y1^2-x2^2-y2^2=0

(M-1)x^2+(M-1)y^2-2(M*x1-2*x2)x-2(M*y1-2*y2)y+(M*x1^2+M*y1^2-x2^2-y2^2)=0

First find the distance between P and A and distance between Pand A' through distance formula.

DISTANCE B/W Pand A:

|PA|= sqrt((x1-x)^2+(y1-y)^2)

=sqrt((x^2-2x*x1+x1^2) +(y^2-2y*y1+y1^2))

|PA'|= sqrt((x2-x)^2+(y2-y)^2)

=sqrt((x^2-2x*x2+x2^2) +(y^2-2y*y2+y2^2))

Now Applying condition:

|PA| = 1

|PA'| M

sqrt((x^2-2x*x1+x1^2) +(y^2-2y*y1+y1^2)) = 1

sqrt((x^2-2x*x2+x2^2) +(y^2-2y*y2+y2^2)) M

Squaring on both sides

(x^2-2x*x1+x1^2) +(y^2-2y*y1+y1^2) = 1

(x^2-2x*x2+x2^2) +(y^2-2y*y2+y2^2) M

Cross Multiplying

M((x^2-2x*x1+x1^2) +(y^2-2y*y1+y1^2)) = (x^2-2x*x2+x2^2) +(y^2-2y*y2+y2^2)

M*x^2-2M*x*x1+M*x1^2+M*y^2-2*M*y*y1+M*y1^2 = (x^2-2x*x2+x2^2) +(y^2-2y*y2+y2^2)

M*x^2-x^2+M*y^2-y^2-2M*x*x1+2x*x2-2*M*y*y1+2y*y2+M*x1^2+M*y1^2-x2^2-y2^2=0

(M-1)x^2+(M-1)y^2-2(M*x1-2*x2)x-2(M*y1-2*y2)y+(M*x1^2+M*y1^2-x2^2-y2^2)=0

asked 2022-07-16

In the diagram to the right, point $P$ lies on the line $y=\frac{3}{2}x$ and is one vertex of square $PQRS$. Point $R$ has coordinates $(5,0)$.

a. Find the coordinates of point $P$.

b. If point $R$ lies on line $l$, and line $I$ divides quadrilateral $PQRS$ into two regions of equal area, find the equation for line $l$.

a. Find the coordinates of point $P$.

b. If point $R$ lies on line $l$, and line $I$ divides quadrilateral $PQRS$ into two regions of equal area, find the equation for line $l$.

asked 2022-08-16

If the points $A(4,3)$ and $B(x,5)$ are on the circle with center $O(2,3)$ find the value of $x$.

Since $AO=BO$ Using distance formula, we get $x=2$

However my approach was why can't section formula be used where $(2,3)$ can be considered as the midpoint so that $M=\frac{{x}_{1}+{x}_{2}}{2}$

Since $AO=BO$ Using distance formula, we get $x=2$

However my approach was why can't section formula be used where $(2,3)$ can be considered as the midpoint so that $M=\frac{{x}_{1}+{x}_{2}}{2}$

asked 2022-08-04

Choose 3 inequalities that form a system whose graph is the shaded region shown above.

$A.y\le -2\phantom{\rule{0ex}{0ex}}B.y\ge -2\phantom{\rule{0ex}{0ex}}C.7x-4y\ge -13\phantom{\rule{0ex}{0ex}}D.7x+2y\ge 17\phantom{\rule{0ex}{0ex}}E.7x-4y\le -13\phantom{\rule{0ex}{0ex}}F.7x+2y\le 17\phantom{\rule{0ex}{0ex}}G.x\ge -2\phantom{\rule{0ex}{0ex}}H.y\le 2$

$A.y\le -2\phantom{\rule{0ex}{0ex}}B.y\ge -2\phantom{\rule{0ex}{0ex}}C.7x-4y\ge -13\phantom{\rule{0ex}{0ex}}D.7x+2y\ge 17\phantom{\rule{0ex}{0ex}}E.7x-4y\le -13\phantom{\rule{0ex}{0ex}}F.7x+2y\le 17\phantom{\rule{0ex}{0ex}}G.x\ge -2\phantom{\rule{0ex}{0ex}}H.y\le 2$

asked 2022-08-03

Compute the hyperbolic length of the following curves joining the points (0,1) and (1,2):

a) y= x + 1

b) y = x^2 +1

c) y= x^3 + 1

a) y= x + 1

b) y = x^2 +1

c) y= x^3 + 1

asked 2022-08-05

Draw the triangle in the plane with vertices (-2,0), (2,0), and (-2,4). What are the y-intercepts?? (Using Moulton Plane)

asked 2022-07-26

Draw a line l and a point P not on l. Construct a perpendicular to l through P. Describe the construction steps and prove the steps are valid (i.e., that the line you construct is perpendicular to L.

asked 2022-07-23

The sphere ${x}^{2}+{y}^{2}+{z}^{2}-2x+6y+14z+3=0$ meets the line joining $A(2,-1,4),B(5,5,5)$ in the points $C$ and $D$. Prove that $AC:CB=-AD:DB=1:2$