Question

find the value of k if tje distance between the point (3,k) and (-3,2) is 2root under 10 unit​

Answer 1

Answer:

k = 0 or 4

Step-by-step explanation:

Given:

• x₁ = 3
• x₂ = -3
• y₁ = k
• y₂ = 2
• Distance = 2√10 units

To find:

Value of k = ?

Solution:

Applying distance formula,

$\bf{Distance = \sqrt{(x _{2} - x_1)^{2} + (y_2 - y_1) ^{2} } }$

Substitute the given values.

$\to \: \sf{Distance = \sqrt{ {( - 3 - 3)}^{2} + {(2 - k)}^{2} } }$

$\to \: \sf{2 \sqrt{10} = \sqrt{ {( - 6)}^{2} + 4 + {k}^{2} - 2(2)k } }$

$\to \: \sf{2 \sqrt{10} = \sqrt{36 + 4 + {k}^{2} - 4k} }$

$\to \: \sf{2 \sqrt{10} = \sqrt{40 + {k}^{2} - 4k} }$

Squaring on both sides,

$\to \: \sf{40 = 40 + {k}^{2} - 4k }$

$\to \: \sf{ \cancel{40} = \cancel{40} + {k}^{2} - 4k }$

$\to \: \sf{ {k}^{2} - 4k = 0}$

$\to \: \sf{k(k - 4) = 0}$

$\therefore \boxed{ \bf{k = 0 \: \: or \: \: 4}}$

Answer 2

Answer:

$\green { \large{\boxed {\sf \therefore \: k = 0 \: \: \: or \: \: \: 4}}}$

Step-by-step explanation:

We know that,

$\sf \: if \: (x_1,y_1) \: \: and \: (x_2, y_2) \: are \: two \: points \\ \sf \: \: then \: distance \: between \: them \: is \: \\ \\ \pink{ \bf \: \sqrt{ {(x_1 - x_2)}^{2} + {(y_1 - y_2)}^{2} } \: \: unit}$

Given that,

Two points are (3,k) and (-3,2)

and diatance between them = $2\sqrt{10}$

According to the question,

$\sf \: x_1 = 3 \\ \sf \: x_2 = - 3 \\ \sf \: y_1 = k \\ \sf \: y_2 = 2$

So,

$\: \: \: \: \sf \: \sqrt{ { \{3 - ( - 3)} \}^{2} + {(k - 2)}^{2} } = 2 \sqrt{10} \\ \\ \sf \implies \: \sqrt{ {(3 + 3)}^{2} + {k}^{2} - 2 \times k \times 2 + {2}^{2} } = 2 \sqrt{10} \\ \\ \sf \implies \: \sqrt{ {6}^{2} + {k}^{2} - 4k + 4} = 2 \sqrt{10} \\ \\ \sf \implies \: \sqrt{36 + {k}^{2} - 4k + 4} = 2 \sqrt{10} \\ \\ \sf \implies \: \sqrt{ {k}^{2} - 4k + 40 } = 2 \sqrt{10} \\ \\ \sf \implies \: {( \sqrt{ {k}^{2} - 4k + 40 } })^{2} = ( {2 \sqrt{10} })^{2} \\ \\ \sf \implies \: {k}^{2} - 4k + 40 = 40 \\ \\ \sf \implies \: {k}^{2} - 4k = 0 \\ \\ \sf \implies \: k(k - 4) = 0 \\ \\ \sf \therefore \: k = 0 \: \: \: \: or \: \: \: \: \: k - 4 = 0 \\ \\ \green {\boxed {\sf \therefore \: k = 0 \: \: \: or \: \: \: 4}}$