Question

⏩Heya,⏪

50 points ✌✌✌
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Please solve this Q25. ASAP:
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Answer for convenience:

(i.) (0,3-4√3)

(ii.) (0,3+4√3)

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▶Note:- PLEASE GIVE ONLY RELEVANT ANSWERS:)

Answer 1

Here is your answer.
24.
Given, P(0, 2) is equidistant from A(3, k) and B(k, 5).

∴ AP = PB

⇒ AP2 = PB2

⇒ (3 – 0)2 + (k – 2)2 = (k – 0)2 + (5 – 2)2    [ Using Distance formula]

⇒ 9 + k2 – 4k + 4 = k2 + 9

⇒ – 4k + 4 = 0

⇒ 4k = 4

⇒ k = 1

Thus, the value of k is 1.
25.
let the given vertices of an equilateral triangle be A(-4,3) and B(4,3).

since y-axis is the perpendicular bisector of AB, therefore point C will lie on y-axis

let the coordinates of the third vertex be C(0,y).
AB2=(-4-4)2+(3-3)2
AB2=8^2+0=64

since ABC is an equilateral triangle.
AC^2=AB^2=BC^2
:.AC^2=AB^2
(0+4)^2+(y-3)^2=64
16+y^2+9-6y=64
y^2-6y -39=0......(1)


solving the quadratic equation (1) by the formula:
in above attachment..


since the origin lies in the interior of the triangle therefore:
in second attachment.

hence the coordinates of the third vertex is

in third attachment..



Hope it helps you.
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Answer 2

❤️❤️❤️❤️ HEY MATE HERE IS YOUR ANSWER ❤️❤️❤️❤️
I)
Let the co-ordinate of third vertex be (x, y) 

Now Using Distance formula 
BC =[4-(-4)]² + (3 - 3)²    
= (4 + 4)² + 0 
BC =√ 8²= 8
 Now , AB = [x - (- 4)]² + (y - 3)²   
  
 AB = (x + 4) 2 + (y - 3) 2 and AC = (x - 4) 2+ (y - 3) 2 
Given, ΔABC is equilateral triangle
∴ AB = AC = BC

Now, AB = AC ⇒ (x + 4) 2 + (y - 3) 2   = (x - 4) 2 + (y - 3) 2

On Squaring both sides, we get

(x + 4)2 + (y – 3)2 = (x – 4)2 + (y – 3)2

(x + 4)2 = (x – 4)2

or x 2 + 16 + 8x = x 2 + 16 – 8x

⇒ 16x = 0

x = 0  ....(1)

AC = BC implies that (x - 4) 2 + (y - 3) 2 = 8(0 - 4) 2 + (y - 3) 2 = 8                [from (1)]

On squaring both sides, we get

16 + y 2 + 9 – 6y = 64

y 2 – 6y – 39 = 0

 y = -(-6) ± (- 6) 2 - 4(1)(-39) 2(1)

 y = 6 ± 36 + 156 2 = 6 ± 192 2

 y =  6 ± 8 3 2 = 3 ± 4 3 

∴ y = 3 + 4√3 and 3 - 4√3 y ≠ 3 + 4 √3 ,
as origin lies in the interior of the triangle. Third vertex = (x, y) = (0, 3 - 4√3).

Hope it's helpful ❤️❤️❤️❤️❤️