Question:

Let $A$ and $B$ be points $ (8,\,\,10) $ and $ (18,\,20), $ respectively. If the point $Q$ divides AB externally in the ratio $ 2:3 $ and $M$ is the $S$ mid-point of $AB$, then the length $MQ $ is equal to

Updated On: Jun 3, 2024
  • $ 25 $
  • $ 5\sqrt{34} $
  • $ 25\sqrt{2} $
  • $ 5\sqrt{26} $
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The Correct Option is C

Solution and Explanation

Given points are $ A(8,\,\,10) $ and $ B(18,\,20) $ . M is the mid-point of AB. Coordinates of $ M=\left( \frac{8+18}{2}.\frac{10+20}{2} \right)=(13,15) $ Point Q divides AB externally in the ration of $ 2:3 $ Day The coordinates of Q
$=\left( \frac{2\times 18-3\times 8}{2-3},\,\frac{2\times 20-3\times 10}{2-3} \right) $
$=\left( \frac{36-24}{-1},\frac{40-30}{-1} \right) $
$=(-12,\,-10) $
Now, length $ MQ=\sqrt{{{(13+12)}^{2}}+{{(15+10)}^{2}}} $
$ \sqrt{{{(25)}^{2}}+{{(25)}^{2}}} $ $ \sqrt{2\times {{(25)}^{2}}} $
$=25\sqrt{2} $
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Concepts Used:

Straight lines

A straight line is a line having the shortest distance between two points. 

A straight line can be represented as an equation in various forms,  as show in the image below:

 

The following are the many forms of the equation of the line that are presented in straight line-

1. Slope – Point Form

Assume P0(x0, y0) is a fixed point on a non-vertical line L with m as its slope. If P (x, y) is an arbitrary point on L, then the point (x, y) lies on the line with slope m through the fixed point (x0, y0) if and only if its coordinates fulfil the equation below.

y – y0 = m (x – x0)

2. Two – Point Form

Let's look at the line. L crosses between two places. P1(x1, y1) and P2(x2, y2)  are general points on L, while P (x, y) is a general point on L. As a result, the three points P1, P2, and P are collinear, and it becomes

The slope of P2P = The slope of P1P2 , i.e.

\(\frac{y-y_1}{x-x_1} = \frac{y_2-y_1}{x_2-x_1}\)

Hence, the equation becomes:

y - y1 =\( \frac{y_2-y_1}{x_2-x_1} (x-x1)\)

3. Slope-Intercept Form

Assume that a line L with slope m intersects the y-axis at a distance c from the origin, and that the distance c is referred to as the line L's y-intercept. As a result, the coordinates of the spot on the y-axis where the line intersects are (0, c). As a result, the slope of the line L is m, and it passes through a fixed point (0, c). The equation of the line L thus obtained from the slope – point form is given by

y – c =m( x - 0 )

As a result, the point (x, y) on the line with slope m and y-intercept c lies on the line, if and only if

y = m x +c