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.
