Question

find a point that is equidistant from the points a(-5,4) b(-1,6) How many such points are there

Answer 1

Answer:

Step-by-step explanation:

Check the answer

Answer 2

Given

Two points $a(-5,4)$ and $b(-1,6)$

To find

A point which is equidistant from both points and how many points are there.

Solution

Midpoint of the two points will be equidistant from the two points.

The mid point will divide the line between the two points in the ratio

$m:n=1:1$

So

$\dfrac{mx_2+nx_1}{m+n},\dfrac{my_2+ny_1}{m+n}\\ =\dfrac{1\times -1+1\times (-5)}{1+1},\dfrac{1\times 6+1\times 4}{1+1}\\ =(-3,5)$

The midpoint of the line joining the two points is (-3,5).is equidistant from a and b.

Finding the equation of the line having the points a and b

$y-6=\dfrac{6-4}{-1+5}(x+1)\\\Rightarrow y=\dfrac{1}{2}x+\dfrac{13}{2}$

Find the equation of the line perpendicular to the line connecting a and b

$m_1\times m_2=-1$ perpendicular line slope product is -1

$\Rightarrow \dfrac{1}{2}m_2=-1\\\Rightarrow m_2=-2$

Equation of line perpendicular to the line joining the points a and b is

$y-5=-2(x+3)\\\Rightarrow y=-2x-6+5\\\Rightarrow y=-2x-1$

Any point on the line $y=-2x-1$ will be equidistant from the points a and b.

So, the number of equidistant points from a and b is infinite.

In the figure the blue line is $y=\dfrac{1}{2}x+\dfrac{13}{2}$ and the green line is $y=-2x-1$