In the xy-plane, a circle C is drawn with center at (1, 2) and radius

To find whether l is tangent or not, we need to prove that l is perpendicular to a line between center and the point where line l touches the circumference of the circle (if at all it does).

For a fact we know that two perpendicular lines have the product of their slope as -1.

Now, to know the slopes we need to know the equations of the line l and the line joining the center and the point where line l touches the circumference of the circle.

One more important thing to remember here is that we can find the equation of the circle using the circle coordinates and the radius. In this case (x-1)^2 + (y-2)^2 = 5^2, although you will find out that we do not need this information on this question.

Now that we are clear what we need to find, lets look at our resources.


1) Point A with coordinates (a, b) lies on line l such that a(a-2) +b(b-4) ≤ 20.

This does not tell us anything about a point where line l meets the circle.
If we dont know whether the line l touches the circle or not, we can not prove anything.
Hence Insufficient.

2)The x-intercept of line l is 10.

Same situation as A. No idea whther the line touches the circle or not.


1 & 2 together -> we still do not have any information about the line l touching the circle or not.

Hence both tother are insufficient, Answer is (E).