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.