Line M is tangent to a circle, which is centered on point

Hi Anaira,

Let me discuss two ways to approach this question.

1. The first (the approach that we personally endorse) is to plot the question on the XY plane. Plotting any coordinate geometry question visually helps us solve the question and bypasses the use of ugly algebraic equations.

We know that the line M is the tangent to the circle with center say C (3,4). This is a Yes/No DS question where we need to see if line M runs through the point (6,6).

Statement 1 : Line M runs through point (-8, 6)

This statement does not give us any information about the radius of the circle. From any external point we can draw two tangents to a circle. So the line M can pass through the point (6,6) and be a tangent at (3,6) or the line M can be a tangent to the circle without passing through the point (6,6). Please refer to the figure below. Insufficient



Statement 2 : Line M is tangent to the circle at point (3, 6)

Since line M is tangent at (3,6), the radius has to be 2 units. We know that a tangent is always perpendicular to the radius at the point of contact, the only tangent possible here will be the line y=6 and this line will always pass through the point (6,6). Refer to the diagram given below. Sufficient



OA : B

2. The second is the algebraic approach which uses the concept of a slope of a line. It is very clear that if the two statements are taken together than we can easily find the slope of the line (0 in this case) and the y intercept here will be 6, so the equations of the line will be y = 6, but this question is a classic C trap, so we need to evaluate the statements individually first before picking option C.

Statement 1 : Just gives us one point (-8,6), to find the slope we require a minimum of two points. Insufficient

Statement 2 : We can find the slope of the line using the points (3,4),the center of the circle, and the point (3,6) the point of tangency. The slope here will be undefined i.e. (6-4)/(3-3) which means that the line is parallel to the y axis. Since we know that the radius and tangent are perpendicular, the slope of the the line M must be parallel to the X axis. The slope of a line parallel to the X axis is 0 and the equation of this line will be y = c. The y intercept c here will be 6 making the equation of line M y = 6. Coordinate (6.6) will definitely lie on the line y = 6. Sufficient.

CrackVerbal Academics Team