Question

CHAPTER: distance formula


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Answer 1

Answer:

distance= speed x time

Answer 2

We know,

$\boxed{ \orange{ \texttt{DISTANCE FORMULA} : \blue{ :\rm \: \sqrt{ {(x_{\tiny{2}} - x_{\tiny{1}})}^{2} + {(y_{\tiny{2}} - y_{\tiny{1}})}^{2} }} } }$

Using the above formula, we have to find the distance between :

1. A(-3, 6) and B(2, -6)
2. P(-a, -b) and Q(a, b)
3. X(3/5, 2) and Y(-1/5, 7/5)
4. S(√3+1, 1) and P(0, √3)

Now,

$\large \bf1. \: \: \text{Measure of AB} \implies\\ \\ \to \sqrt{ { \{6 - ( - 3) \}}^{2} + { \{ - 6 - 2 \}}^{2} } \\ \to \sqrt{ {(6 + 3)}^{2} + {( - 8)}^{2} } \\ \to \sqrt{ {(9)}^{2} + 64 } \: \: = \sqrt{81 + 64} \\ \to \rm \sqrt{145} \: \: \: or \: \: \: \red{12.04} \: units$

$\overline{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: }$

$\large \bf2. \: \: \text{Measure of PQ} \implies \\ \\ \to \tt \sqrt{ { \{ - a - ( - b) \}}^{2} + {(b - a)}^{2} } \\ \to \tt \sqrt{ { \{ - a + b\}}^{2} + {(b - a)}^{2} } \\ \to \tt \sqrt{ {(b - a)}^{2} + \{ {b}^{2} + {a}^{2} - 2ba \} } \\ \to \tt \sqrt{ {b}^{2} + {a}^{2} - 2ba + {b}^{2} + {a}^{2} - 2ba } \\ \to \tt \sqrt{ {2b}^{2} + {2a}^{2} - 4ba } \\ \to \tt \sqrt{2( {b}^{2} + {a}^{2} - 2ba) } \\ \to \tt\big( \sqrt{2} \big ) \bigg( \sqrt{ {(b - a)}^{2} } \bigg) \\ \to \tt \red{\sqrt{2} \big(b - a\big)} \: \: units$

$\overline{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: }$

$\large \bf3. \: \: \: \text{Measure of XY} \implies \\ \\ \to \rm \sqrt{ {\bigg(2 - \frac{3}{5} \bigg )}^{2} + {\bigg \{ \frac{7}{5} - \bigg( \frac{ - 1}{5} \bigg) \bigg \}}^{2} } \\ \\ \to \sqrt{ {\bigg( \frac{10 - 3}{5} \bigg)}^{2} + { \bigg\{ \frac{7 - ( - 1)}{5} \bigg\}}^{2} } \\ \\ \to \sqrt{ {\bigg( \frac{7}{5} \bigg)}^{2} + {\bigg( \frac{8}{5} \bigg)}^{2} } \\ \\ \to\sqrt{ {\bigg( \frac{49}{25} \bigg)}^{} + \bigg( \frac{64}{25} \bigg) } \\ \\ \to \sqrt{ \frac{49 + 64}{25} } \: \: = \sqrt{ \frac{113}{25} } \\ \\ \rm \longrightarrow \red{ \frac{ \sqrt{113} }{5} } \: \: units$

$\overline{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: }$

$\large \bf4 . \: \: \text{Measure of SP} \implies \\ \\ \to \sqrt{ { \{1 - ( \sqrt{3} + 1 ) \}}^{2} + {( \sqrt{3} - 0 )}^{2} } \\ \\ \to \sqrt{ { \{1 - \sqrt{3} - 1 \}}^{2} + {( \sqrt{3} )}^{2} } \\ \\ \to \sqrt{ {( - \sqrt{3} )}^{2} + 3} \\ \\ \to \sqrt{3 + 3} \: \: \: = \rm \red{\sqrt{6}} \: \: units$

✓✓ Extra Formula

$\texttt{Section Formula} : \sf \bigg \{ \frac{m_1 x_2 +m_2 x_1 }{m_1 +m_2 } , \frac{m_1 y_2 +m_2 y_1 }{m_1 +m_2} \bigg \}$