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100 points for $49 worth of Veritas practice GMATs FREE VERITAS PRACTICE GMAT EXAMS Earn 10 Points Per Post Earn 10 Points Per Thanks Earn 10 Points Per Upvote ## Circle ABCD in the diagram above is defined by the equation ##### This topic has 2 expert replies and 0 member replies ### Top Member ## Circle ABCD in the diagram above is defined by the equation ## Timer 00:00 ## Your Answer A B C D E ## Global Stats Difficult Circle ABCD in the diagram above is defined by the equation x^2+y^2=25. Line segment EF is defined by the equation 3y=4x+25 and is tangent to circle ABCD at exactly one point. What is the point of tangency? A. (-4, 3) B. (-3, 4) C. (-4, 7/2) D. (-7/2, 3) E. (-4, 4) OA A Source: Veritas Prep ### GMAT/MBA Expert GMAT Instructor Joined 25 May 2010 Posted: 15348 messages Followed by: 1864 members Upvotes: 13060 GMAT Score: 790 Top Reply BTGmoderatorDC wrote: Circle ABCD in the diagram above is defined by the equation x^2+y^2=25. Line segment EF is defined by the equation 3y=4x+25 and is tangent to circle ABCD at exactly one point. What is the point of tangency? A. (-4, 3) B. (-3, 4) C. (-4, 7/2) D. (-7/2, 3) E. (-4, 4) Circle ABCD: The equation for a circle with a center at the origin and a radius of r is xÂ² + yÂ² = rÂ². The equation for circle ABCD is xÂ² + yÂ² = 5Â². Thus, circle ABCD has a radius of 5. Line EF: 3y = 4x + 25 y = (4/3)x + 25/3 Slope = 4/3. A radius drawn to a point of tangency forms a RIGHT angle with the tangent line and thus is PERPENDICULAR to the tangent line. Draw a radius to the point at which line EF is tangent to circle ABCD: The dashed line forming the radius is perpendicular to line EF. The slopes of perpendicular lines are NEGATIVE RECIPROCALS. Since the slope of line EF = 4/3, the slope of the dashed line forming the radius = -(3/4). Implication: To travel along the dashed line from the origin to the point of tangency, we must move to the left 4 places and up 3 places, yielding the 3-4-5 triangle shown in the figure above. Thus, the point of tangency = (-4, 3). The correct answer is A. _________________ Mitch Hunt Private Tutor for the GMAT and GRE GMATGuruNY@gmail.com If you find one of my posts helpful, please take a moment to click on the "UPVOTE" icon. Available for tutoring in NYC and long-distance. For more information, please email me at GMATGuruNY@gmail.com. Student Review #1 Student Review #2 Student Review #3 Free GMAT Practice Test How can you improve your test score if you don't know your baseline score? Take a free online practice exam. Get started on achieving your dream score today! Sign up now. ### GMAT/MBA Expert GMAT Instructor Joined 09 Oct 2010 Posted: 1449 messages Followed by: 32 members Upvotes: 59 Top Reply BTGmoderatorDC wrote: Circle ABCD in the diagram above is defined by the equation x^2+y^2=25. Line segment EF is defined by the equation 3y=4x+25 and is tangent to circle ABCD at exactly one point. What is the point of tangency? A. (-4, 3) B. (-3, 4) C. (-4, 7/2) D. (-7/2, 3) E. (-4, 4) Source: Veritas Prep Obs.: when a given line is tangent to a given circle, the point of tangency is (by definition) unique. Therefore the part in red in the question stem could (and should) be omitted! $$\left( * \right)\,\,\left\{ \matrix{ \,\left( {\rm{I}} \right)\,\,\,{x_p}^2 + {y_p}^2 = 25 \hfill \cr \,\left( {{\rm{II}}} \right)\,\,3{y_p} = 4{x_p} + 25 \hfill \cr} \right.$$ $$? = \left( {{x_p},{y_p}} \right)\,\,\,\,\,\,\,\left[ {\,p = {\rm{particular}}\,} \right]$$ We believe testing the alternative choices is much faster than solving the system above, hence: $$\left( A \right)\,\,\,\left( {{x_p},{y_p}} \right) = \left( { - 4,3} \right)\,\,\,{\rm{satisfies}}\,\,\left( * \right)\,\,?$$ $$\left\{ \matrix{ \left( {\rm{I}} \right)\,\,\,{\left( { - 4} \right)^2} + {\left( 3 \right)^2}\,\mathop = \limits^? \,\,25\,\,\,\,\,\,\,\, \Rightarrow \,\,\,{\rm{yes}}! \hfill \cr \left( {{\rm{II}}} \right)\,\,\,3\left( 3 \right)\,\,\mathop = \limits^? \,\,\,4\left( { - 4} \right) + 25\,\,\,\, \Rightarrow \,\,\,{\rm{yes}}! \hfill \cr} \right.\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,? = \left( { - 4,3} \right)\,\,\,\, \Rightarrow \,\,\,\,\,\left( A \right)$$ This solution follows the notations and rationale taught in the GMATH method. Regards, Fabio. _________________ Fabio Skilnik :: GMATH method creator ( Math for the GMAT) English-speakers :: https://www.gmath.net Portuguese-speakers :: https://www.gmath.com.br • Get 300+ Practice Questions 25 Video lessons and 6 Webinars for FREE Available with Beat the GMAT members only code • 1 Hour Free BEAT THE GMAT EXCLUSIVE Available with Beat the GMAT members only code • Free Veritas GMAT Class Experience Lesson 1 Live Free Available with Beat the GMAT members only code • Magoosh Study with Magoosh GMAT prep Available with Beat the GMAT members only code • 5-Day Free Trial 5-day free, full-access trial TTP Quant Available with Beat the GMAT members only code • Award-winning private GMAT tutoring Register now and save up to$200

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