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x^2 + y^2 -8x + 4y = 30 &

y = x + 4

Where the tangent touches The (x,y) values will be the same.

So squaring the linear equation and substituting.

y^2 = (x + 4)^2

Hence

x^2 + (x + 4)^2 - 8x + 4(x + 4) = 30

Multiply out

x^2 + x^2 + 8x + 16 -8x + 4 x + 16 = 30

Collect 'like terms'.

2x^2 + 4x + 32 = 30

2x^2 + 4x + 2 = 0

Divide by '2'

x^2 + 2x + 1 = 0

Factor (x +1)^2 = 0

x = -1

When x = -1 y = 3 This is the point of contact of the tangent with the circumference.

Since x = -8x , then the x-co-ord of the centre is '4' and y = 4y , y-co-ord of the centre is '-2'.

In (x,y) tangent point is (-1,3) and circle centre is (4,-2). The radius is the distance between these two points.

By Pythagoras.

(-1 - 4)^2 + (3 - - 2)^2 = r^2

(-5)^2 + (5)^2 = r^2

25 + 25 = r^2

50 = r^2

r = sqrt(50) = sqrt(2 x 25) = 5sqrt(2)

Done !!!! Hope that helps!!!

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lenpollock

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1y ago
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6y ago

Centre of circle: (4, -2)

Slope of tangent: 1

Slope of radius: -1

Radius equation: y--2 = -1(x-4) => y = -x+2

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6y ago

The equation for the radius which intersects the tangent is x + y = 2.

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Q: What is the radius equation of the circle x2 plus y2 -8x plus 4y equals 30 when touched by the tangent line y equals x plus 4?
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