Question

(ii) If the distance between the points (x, -1) and (3, 2) is 5, then the value of x is *
(a) -7 or -1
O (b) -7 or 1
O (c) 7 or 1
(d) 7 or -1​

Answer 1

Answer:

Given :-

• The distance between the points (x, - 1) and (3, 2) is 5.

To Find :-

• What is the value of x.

Formula Used :-

$\clubsuit$ Distance Formula :-

$\small\mapsto \sf\boxed{\bold{\pink{Distance =\: \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}}}}$

where,

• (x₁ , y₁) = Co-ordinates of the first points
• (x₂ , y₂) = Co-ordinates of the second points

Solution :-

Given :-

• Distance = 5

Given Points :

➲ (x , - 1)

➲ (3 , 2)

where,

• x₁ = x
• y₁ = - 1
• x₂ = 3
• y₂ = 2

According to the question by using the formula we get,

$\implies \sf 5 = \sqrt{(3 - x)^2 + \{2 - (- 1)\}^2}$

$\implies \sf 5 =\: \sqrt{(3 - x)^2 + (2 + 1)^2}$

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

$\implies \sf (5)^2 =\: (3 - x)^2 + 3^2$

$\implies \sf 25 =\: 9 + x^2 - 6x + 9$

$\implies \sf 25 =\: x^2 - 6x + 9 + 9$

$\implies \sf 25 =\: x^2 - 6x + 18$

$\implies \sf 25 - 18 =\: x^2 - 6x$

$\implies \sf 7 =\: x^2 - 6x$

$\implies \sf x^2 - 6x - 7 =\: 0$

$\implies \sf x^2 - (7 - 1)x - 7 =\: 0$

$\implies \sf x^2 - 7x + x - 7 =\: 0\: \: \bigg\lgroup \sf\bold{\pink{By\: doing\: middle\: term}}\bigg\rgroup$

$\implies \sf x(x - 7) + 1(x - 7) =\: 0$

$\implies \sf (x + 1)(x - 7) =\: 0$

$\implies \bf (x + 1) =\: 0$

$\implies \sf\bold{\red{x =\: - 1}}$

$\implies \bf (x - 7) =\: 0$

$\implies \sf\bold{\red{x =\: 7}}$

${\small{\bold{\underline{\therefore\: The\: value\: of\: x\: is\: - 1\: or\: 7\: .}}}}$

Hence, the correct options is option no d) 7 or - 1.

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★ EXTRA INFORMATION ★

❒ Section Formula :

$\mapsto \sf\boxed{\bold{\red{\bigg\lgroup \dfrac{m_1x_2 + m_2x_1}{m_1 + m_2} , \dfrac{m_1y_2 + m_2y_1}{m_1 + m_2}\bigg\rgroup}}}$

❒ Mid-Point Formula :

$\mapsto \sf\boxed{\bold{\pink{\bigg\lgroup \dfrac{x_1 + x_2}{2} , \dfrac{y_1 + y_2}{2}\bigg\rgroup}}}$

❒ Centroid Formula :

$\mapsto \sf\boxed{\bold{\green{\bigg\lgroup \dfrac{x_1 + x_2 + x_3}{3} , \dfrac{y_1 + y_2 + y_3}{3}\bigg\rgroup}}}$

Answer 2

Given: The distance b/w the two given points (x, – 1) and (3, 2) is 5.

Need to find: The value of x?

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✇ To find out distance b/w any two points the require formula is Given by —

$\star\:\underline{\boxed{\pmb{\sf{Distance = \sqrt{\bigg(x_2 - x_1\bigg)^2 + \bigg(y_2 - y_1\bigg)^2}}}}}$

where,

• x₁ = x

• x₂ = 3

• y₁ = – 1

• y₂ = 2

⠀

⠀⠀⠀$\underline{\bf{\dag} \:\mathfrak{Putting\;these\;values\;in\;formula :}}\\\\:\implies\sf \sqrt{\Big(x - 3\Big)^2 + \Big(-1 - 2\Big)^2} = 5 \\\\\\:\implies\sf \sqrt{\Big(x - 3\Big)^2 + 3^2} = 5^2\\\\\\:\implies\sf \Big(x - 3\Big)^2 + 9 = 5^2\\\\\\:\implies\sf x^2 - 6x + 18 = 25\\\\\\:\implies\sf x^2 - 6x = 25 - 18\\\\\\:\implies\sf x^2 - 6x -7 = 0 \\\\\\:\implies\sf x^2 - 7x + x - 7 = 0\\\\\\:\implies\sf x(x - 7) +1(x - 7) = 0\\\\\\:\implies\underline{\boxed{\pmb{\frak{x = 7\;\&\;-1}}}}\;\bigstar$

⠀

$\therefore\:{\underline{\sf{Hence\;the\; value\;of\:x\;is\; {\textsf{\textbf{Option d) 7 or -1}}}.}}}$

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$\quad\qquad\boxed{\underline{\underline{\bigstar \: \bf\:More \;to\;know\:\bigstar}}}$⠀⠀⠀

¤ Another formula to calculate the co–ordintes of the point dividing the line segment joining two points (x₁, x₂ ) and (y₁, y₂) in the ratio of m₁ : m₂ is Section formula. It is given as:

• $\Bigg(\sf\dfrac{m_1x_2+m_2x_1}{m_1+m_2},\;\dfrac{m_1y_2+m_2y_1}{m_1+m_2}\Bigg)$