Find a point which is equidistant from the points A (–5, 4) and B (–1, 6)? How many such points are there

Solution:

Given, the points are A(-5, 4) and B(-1, 6)

We have to find a point which is equidistant from the given points.

Let the point P(x, y) be equidistant from the points A(-5, 4) and B(-1, 6)

So, PA = PB

The distance between two points P (x₁ , y₁) and Q (x₂ , y₂) is

√[(x₂ - x₁)² + (y₂ - y₁)²]

Distance between P(x, y) and A(-5, 4) = √[(-5 - x)² + (4 - y)²]

Distance between P(x, y) and B(-1, 6) = √[(-1 - x)² + (6 - y)²]

Now, √[(-5 - x)² + (4 - y)²] = √[(-1 - x)² + (6 - y)²]

On squaring both sides,

(-5 - x)² + (4 - y)² = (-1 - x)² + (6 - y)²

By using algebraic identity,

(a + b)² = a² + 2ab + b²

(a - b)² = a² - 2ab + b²

25 + 10x + x² + y² - 8y + 16 = x² + 2x + 1 + y² - 12y + 36

x² + y² + 10x - 8y + 41 = x² + y² + 2x - 12y + 37

Canceling out common terms,

10x - 8y + 41 = 2x - 12y + 37

By grouping,

10x - 2x - 8y + 12y + 41 - 37 = 0

8x + 4y + 4 = 0

Dividing by 4,

2x + y + 1 = 0 ---------------- (1)

The coordinates of the mid-point of the line segment joining the points P (x₁ , y₁) and Q (x₂ , y₂) are [(x₁ + x₂)/2, (y₁ + y₂)/2]

Midpoint of A(-5, 4) and B(-1, 6) = [(-5 + (-1))/2, (6 + 4)/2]

= [-6/2, 10/2]

= (-3, 5)

Verification:

Substitute (-3, 5) in (1)

So, 2(-3) + 5 + 1 = 0

-6 + 6 = 0

It is clear that the midpoint of AB satisfies the equation (1).

The solution of the equation 2x + y + 1 = 0 are all equidistant from the points A and B.

Therefore, there are an infinite number of points that are equidistant from the points A and B.

✦ Try This: If the point A(2, -4) is equidistant from P(3, 8) and Q(-10, y), then find the value of y. Also, find distance PQ.

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 7

NCERT Exemplar Class 10 Maths Exercise 7.3 Problem 5

Find a point which is equidistant from the points A (–5, 4) and B (–1, 6)? How many such points are there

Summary:

The solution of the equation 2x+y+1=0 is equidistant from the points A (–5, 4) and B (–1, 6). There are an infinite number of points that are equidistant from the points A and B

☛ Related Questions: