Line l is tangent to a circle whose centre is at (3, 2).

Question

Line l is tangent to a circle whose centre is at (3, 2). If the point of tangency is (6, 6), what is the slope of line l ?

A. -4/3
B. -3/4
C. 0
D. 3/4
E. 4/3

Hi, can someone please help me explain the solution of the above question. According to the solution set that i have, the ans is A, but i am not sure how. Thank you.

Bunuel · Accepted Answer

The tangent line is always at the 90 degree angle (perpendicular) to the radius of a circle (to the point of tangency). Untitled.png Thus that radius passes through the points (3, 2) and (6, 6). Its slope is, rise over run, \frac{y_2-y_1}{x_2-x_1}=\frac{6-2}{6-3}=\frac{4}{3} . The line perpendicular to that radius has the slope which is negative reciprocal of 4/3, thus -3/4. Answer: B. For more check: math-circles-87957.html math-coordinate-geometry-87652.html Hope this helps.

Nunuboy1994 · Answer

There's more the one way to solve this question; however, within the scope of realistic GMAT practice- the most simple way is to just use the slope formula and take the negative reciprocal. Why? What coordinates should be used for the slope formula? Firstly, this question tells us that the point of tangency lies on the coordinate (6,6) - what this simply means is that at this coordinate is a point that part of both on the circle AND on the line that touches it. If we plug both coordinates into the slope equation

y2-y1/ x2-x1= this would equal the slope of the circle's radius

According to geometric principles, if we take the negative reciprocal of the slope of the radius just like how we take the negative reciprocal of a parallel line to find the slope of its perpendicular line then we would have the line of the tangent line.

Hence B

ThatDudeKnows · Answer

For the vast majority of slope questions (of all difficulty levels!), you don't need to actually calculate the slope. Plot the two points cited in the question stem. Draw a line through them. Now draw a line perpendicular to that one. Does that second line go up and to the right or down and to the right. Down. That means the slope is negative. C, D, and E are out. A line with a slope of 45 degrees has a slope of +/- 1. If it is steeper than 45 degrees, the slope is greater than 1 if positive or less than -1 if negative. If it is less steep than 45 degrees, the slope is between 0 and 1 if positive or between 0 and -1 if negative. Is our line steeper than 45 degrees or less steep than 45 degrees? Less steep. We want something between 0 and -1. We did not have to actually calculate the slope in order to get the question right in a matter of seconds and with basically zero chance of making a silly mistake (like we might have if we'd done the math).

Answer choice B.

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Line l is tangent to a circle whose centre is at (3, 2).

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Line l is tangent to a circle whose centre is at (3, 2).

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

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My Rewards