Question

The distance between the points (4,p) and (1,0) is 5 then find the value of p​

Answer 1

Given:

• The distance between the points (4,p) and (1,0) is 5.

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To find:

• Value of p?

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Solution:

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☯ Let (4,p) and (1,0) be the cordinate of points A and B respectively.

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$\setlength{\unitlength}{14mm}\begin{picture}(7,5)(0,0)\thicklines\put(0,0){\line(1,0){5}}\put(5.1, - 0.3){\sf B}\put( - 0.2, - 0.3){\sf A}\put(5.2, 0){\sf (1,0)}\put( - 0.7, 0){\sf (4,p)}\end{picture}$

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$\bigstar\:{\underline{\sf{\purple{Using\;Distance\;formula\;:}}}}$

$\star\;{\boxed{\sf{\pink{d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}}}}}$

$\sf Here \begin{cases} & \sf{x_1 , y_1 = 4, p} \\ & \sf{x_2 , y_2 = 1, 0} \end{cases}$

$\dag\;{\underline{\frak{Putting\;values,}}}$

$:\implies\sf 5 = \sqrt{(1 - 4)^2 + (0 - p)^2}$

$:\implies\sf 5 = \sqrt{(-3)^2 + (p)^2}$

$:\implies\sf 5 = \sqrt{9 + p^2}$

$:\implies\sf (5)^2 = \sqrt{(9 + p^2)^2}\qquad\quad\bigg\lgroup\bf Squaring\;both\;sides \bigg\rgroup$

$:\implies\sf 25 = 9 + p^2$

$:\implies\sf p^2 = 25 - 9$

$:\implies\sf p^2 = 16$

$:\implies\sf \sqrt{p^2} = \sqrt{16}$

$:\implies{\underline{\boxed{\sf{\purple{p = +4, -4}}}}}\;\bigstar$

$\therefore\;{\underline{\sf{Value\;of\;p\;is\; {\textsf{\textbf{+4\;or\;-4}}}.}}}$

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$\qquad\quad\boxed{\bf{\mid{\overline{\underline{\pink{\bigstar\:More\:to\:know :}}}}}\mid}$

• Section Formula: The section formula tells us the coordinates of the point which divides a given line segment into two parts such that their lengths are in the ratio m : n.

• $\sf M = \bigg( \dfrac{m x_2 + n x_1}{m + n}\;,\; \dfrac{m y_2 + n y_1}{m + n} \bigg)$

Answer 2

Question -

The distance between the points (4,p) and (1,0) is 5 then find the value of p.

Answer -

Given that -

• The distance between the points (4,p) and (1,0) is 5.

To find -

• Value of p.

Solution -

• Value of p = +4 or -4

Assumptions -

• Point (4,p) and (1,0) be X and Y respectively.

Using formula

• Distance formula =

$d = \sqrt{(x _{2} - x _{1}} ) ^{2} + {(y _{2} - y _{1}} ) ^{2}$

Let's understand the concept 1st

• This question says that 5 is the distance between two points (4,p) and (1,0). Afterwards it ask us to find the value of p.

How to solve this question -

• To solve this question we have to use the formula of distance. Putting the values according to formula. We get our final result that is -4 or +4

Full solution -

Let point (4,p) and (1,0) be X and Y respectively.

$\setlength{\unitlength}{14mm}\begin{picture}(7,5)(0,0)\thicklines\put(0,0){\line(1,0){5}}\put(5.1, - 0.3){\bf Y}\put( - 0.2, - 0.3){\bf X}\put(5.2, 0){\tt (1,0)}\put( - 0.7, 0){\tt (4,p)}\end{picture}$

Using distance formula =

$d = \sqrt{(x _{2} - x _{1}} ) ^{2} + {(y _{2} - y _{1}} ) ^{2}$

Here,

• x₁ , y₁ = 4,p (cordinate of 1st point)

• x₂ , y₂ = 1,0 (cordinate of 2nd point)

Now, putting the values according to formula we get the following results.

$\mapsto$ $5 = \sqrt{(1 - 4) ^{2} + (0 - p) ^{2} }$

$\mapsto$ $5 = \sqrt{( - 3) ^{2} + ({p}^{2}) }$

$\mapsto$ $5 = \sqrt{9 + {p}^{2} }$

$\mapsto$ $( {5}^{2} ) = \sqrt{(9 + {p}^{2} ) ^{2} }$

$\mapsto$ 25 = 9 + p²

$\mapsto$ p² = 25 - 9

$\mapsto$ p² = 16

$\mapsto$ p = √16

$\mapsto$ p = 4

$\mapsto$ p = +4 or -4

Hence, the value of p = +4 or -4