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3.a circle of radius 5 units passes through the points (- 3,3) and (3,1) .
i.how many circles can be drawn that meet the given criteria?
ii. what can be the centre of a circle(S)
b. the equation of any two diameters of a circle are given by the straight lines x-y-4=0 and 2x+ 3y + 7 =0.
find the equation of the circle if it passes through the point (2,4).​

3a Circle Of Radius 5 Units Passes Through The Points 33 And 31 Ihow Many Circles Can Be Drawn That Meet The Given Criteriaii What Can Be The Centre Of A Circle class=

Sagot :

mhanifa

Answer:

  • See below

Step-by-step explanation:

Given:

  • Points (- 3,3) and (3,1) on a circle
  • r = 5

i.

There are possible two points that can have a distance of 5 units from both of the given points, so possible two centers, hence two possible circles.

ii.

Let the points are A and B and the centers of circles are F and G.

The midpoint of AB, the point C is:

  • C = ((-3 + 3)/2, (3 + 1)/2) = (0, 2)

The length of AB:

  • AB = [tex]\sqrt{(3 + 3)^2 + (1 - 3)^2} = \sqrt{6^2+2^2} = \sqrt{40} = 2\sqrt{10}[/tex]

The distance AC = BC = 1/2AB = [tex]\sqrt{10}[/tex]

The distance FC or GC is:

  • FC = GC = [tex]\sqrt{5^2-10} = \sqrt{15}[/tex]

Possible coordinates of center are (h, k).

We have radius:

  • (h + 3)² + (k - 3)² = 25
  • (h - 3)² + (k - 1)² = 25

Comparing the two we get:

  • (h + 3)² + (k - 3)² = (h - 3)² + (k - 1)²

Simplifying to get:

  • k = 3h + 2

We consider this in the distance FC or GC:

  • h² + (k - 2)² = 15
  • h² + (3h + 2 - 2)² = 15
  • 10h² = 15
  • h² = 1.5
  • h = √1.5 or
  • h = - √1.5

Then k is:

  • k = 3√1.5 + 2 or
  • k = -3√1.5 + 2

So coordinates of centers:

  • (√1.5, 3√1.5 + 2) for G or
  • (√1.5, 3√1.5 + 2) for F (or vice versa)

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b.

Diameters:

  • x - y - 4 = 0
  • 2x + 3y + 7 = 0

The intercession of the diameters is the center. We solve the system above and get. Not solving here as it is already a long answer:

  • x = 1, y = -3

The point (2, 4) on he circle given.

Find the radius which is the distance between center and the given point:

  • r = [tex]\sqrt{(2 - 1)^2+(4+3)^2} = \sqrt{1^2+7^2} = \sqrt{50}[/tex]

The equation of circle:

  • (x - 1)² + (y + 3)² = 50
View image mhanifa
View image mhanifa
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