need help to this problem

Question

A circle passes through (-2,4) and touches the y-axis at (0,2). Which one of the following equations can represent a diameter of this circle :-               

 

(a) 4x+5y-6=0

(b) 2x-3y+10=0

(c) 3x+4y-3=0

(d) 5x+2y+4=0

Answer 1

Hello Akasapu,

 

The key phrase here is that the circle touches the y-axis at (0, 2). Mathematically, that means that the y-axis is the tangent line for the circle at (0, 2).

 

Now, strictly from a mathematical standpoint, this phrase could also mean that the circle intersects the y-axis at (0, 2). If we broaden our definition of touches to mean intersect, then this problem becomes a little bit longer, because we can no longer assume that the y-axis is tangent to the circle at (0, 2). Fortunately, we can find a solution without worrying about this technicality.

 

So, if the y-axis is tangent to the circle at (0, 2), then the center of the circle lies somewhere along the line y = 2.

 

By a visual inspection after plotting the points (-2, 4) and (0, 2), we can see that the center of the circle must be at (-2, 2), since (-2, 2) is equidistant to both of the points (-2, 4) and (0, 2).

 

Well, any diameter of a circle must pass through the center of that circle, so one of the four lines presented above must pass through the point (-2, 2).

 

Hope this has helped! :)