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A line intersects the y-axis and x-axis at the points P and Q respectively.If (2,-5) is the midpoint of PQ, then find the coordinates of P and Q.


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Answers

Answered by mathdude500
47

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

Given that,

  • A line intersects the y-axis and x-axis at the points P and Q respectively and (2,-5) is the midpoint of PQ.

Let assume that

  • Coordinates of point P be (0, a)

  • Coordinates of point Q be (b, 0)

Now,

  • (2, - 5) is the midpoint of PQ.

We know

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:

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

So, using Midpoint Formula, we get

\red{\rm :\longmapsto\:(2, - 5) = \bigg(\dfrac{0 + b}{2}, \: \dfrac{a + 0}{2}  \bigg) }

\red{\rm :\longmapsto\:(2, - 5) = \bigg(\dfrac{b}{2}, \: \dfrac{a}{2}  \bigg) }

So, on comparing, we get

\rm :\longmapsto\:\dfrac{b}{2}  =2 \:  \:  \: and \:  \:  \:  \dfrac{a}{2}  =  - 5

\bf\implies \:b = 4 \:  \: and \:  \: a =  - 10

So,

  • Coordinates of point P be (0, - 10)

  • Coordinates of point Q be (4, 0)

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More to know :-

1. Section formula.

Let P(x₁, y₁) and Q(x₂, y₂) be two points in the coordinate plane and R(x, y) be the point which divides PQ internally in the ratio m₁ : m₂. Then, the coordinates of R will be:

\boxed{\tt{ \sf\implies R = \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. Distance formula.

Let P(x₁, y₁) and Q(x₂, y₂) be two points in the coordinate plane, then distance between P and Q is

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

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:

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

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