Question

Solve this solve this​

Answer 1

$\large\underline{\sf{Solution-}}$

Given that

• Coordinates of P be (x, y)

• Coordinates of A be (6, 1)

• Coordinates of B be (1, 6)

We know,

Distance Formula :- Distance between two points is calculated by using the formula given below,

Let A(x₁, y₁) and B(x₂, y₂) be two points in the coordinate plane, then distance between A and B is given by

$\boxed{ \tt{ \: AB \: = \: \sqrt{ {(x_{1} - x_{2}) }^{2} + {(y_{2} - y_{1})}^{2} }}}$

Further given that,

P is equidistant from A and B.

$\purple{\rm :\longmapsto\:PA=PB}$

$\purple{\rm :\longmapsto\:PA^{2} =PB ^{2} }$

$\purple{\rm :\longmapsto\: {(x - 6)}^{2} + {(y - 1)}^{2} = {(x - 1)}^{2} + {(y - 6)}^{2}}$

$\purple{\rm :\longmapsto\: {x}^{2} + 36 - 12x + {y}^{2} + 1 - 2y = } \\ \\ \purple{ \rm \: {x}^{2} + 1 - 2x + {y}^{2} + 36 - 12y}$

$\purple{\rm :\longmapsto\:37 - 12x - 2y \: = \: 37 - 12y - 2x}$

$\purple{\rm :\longmapsto\: - 12x - 2y \: = \: - 12y - 2x}$

$\purple{\rm :\longmapsto\: - 12x + 2x \: = \: - 12y + 2y}$

$\purple{\rm :\longmapsto\: - 10x \: = \: - 10y}$

$\purple{\rm :\longmapsto\: x \: = \: y}$

Hence,

$\purple{\rm \implies\:\boxed{ \tt{ \: x \: - \: y \: = \: 0 \: }}}$

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Explore More :-

1. Section formula

Let A(x₁, y₁) and B(x₂, y₂) be two points in the coordinate plane and C(x, y) be the point which divides AB internally in the ratio m₁ : m₂. Then, the coordinates of C will be

$\sf\implies C = \bigg(\dfrac{m_{1}x_{2}+m_{2}x_{1}}{m_{1}+m_{2}}, \dfrac{m_{1}y_{2}+m_{2}y_{1}}{m_{1}+m_{2}}\bigg)$

2. Mid-point formula

Let P(x₁, y₁) and Q(x₂, y₂) be two points in the coordinate plane and R(x, y) be the mid-point of PQ. Then, the coordinates of R will be:

$\sf\implies R = \bigg(\dfrac{x_{1}+x_{2}}{2}, \dfrac{y_{1}+y_{2}}{2}\bigg)$

3. Centroid of a triangle

Centroid of a triangle is the point where the medians of the triangle meet.

Let A(x₁, y₁), B(x₂, y₂) and C(x₃, y₃) be the vertices of a triangle. Let R(x, y) be the centroid of the triangle. Then, the coordinates of R will be:

$\sf\implies R = \bigg(\dfrac{x_{1}+x_{2}+x_{3}}{3}, \dfrac{y_{1}+y_{2}+y_{3}}{3}\bigg)$