Bunuel wrote:

rxs0005 wrote:

In the XY co-ordinate plane , circle C has center at ( 8,0 ) and tangent to the line y = x

what is the diameter of the circle

8

4 * root(2)

8 * root(2)

16

16 * root(2)

Refer to the diagram below:

Attachment:

1.PNG

If a line is tangent to a circle, then a radius drawn to the point of contact is perpendicular to that line.

Now, as the line y=x makes 45 degrees with the axis then we have 45-45-90 right triangle with hypotenuse equal to 8, thus the leg/radius (red segment) equals to \(\frac{8}{\sqrt{2}}\) and the diameter equals to \(2*\frac{8}{\sqrt{2}}=8*\sqrt{2}\).

Answer: C.

Other method to solve such questions. It's slightly lengthy in the beginning but kinda foolproof

The point at which line y=x touches the circle is perpendicular to the circle.

Slope of line y=x is 1

Thus slope of the line (centre of the circle) touching the point of tangent is -1 & its equation will be (y-0)/(x-8)= -1

y = -x +8..............(1)

y = x ...................(2)

Point of intersection of line (1) & (2) is

x = -x +8

2x = 8 ----> x =4

The other co-ordinate of point of intersection will be

y = -x +8---> y = -4+8---->y=4

The co-ordinates of the point of intersection is (4,4)

Now the distance between point (4,4) & (8,0) will be the radius of the circle

Radius = √((0-4)^2+(8-4)^2 )=4√2

Therefore diameter = 8√2

Answer: C

Hope it will help many others to come.

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