question_answer

If the ends of the base of an isosceles triangle are at (2, 0) and (0, 1) and the equation of one side is x = 2, then the orthocenter of the triangle is

A) (3/2, 3/2)

B)  (5/4, 1)

C)  (3/4, 1)

D)  (4/3, 7/12)

Correct Answer: B

Solution :

From the figure, \[{{2}^{2}}+{{({{y}_{1}}-1)}^{2}}=y_{1}^{2}\]

\[4+y_{1}^{2}+1-2{{y}_{1}}=y_{1}^{2}\]

\[5=2{{y}_{1}}\]or \[{{y}_{1}}=5/2\]

Equation of the line from (2, 5 / 2) to the given base is

\[y-5/2=2\,\,(x-2)\]

\[\Rightarrow \]   \[2y-5=4\,\,(x-2)y=1,\]

\[\Rightarrow \]   \[-\,\,3/4=x-2\]

\[\therefore \]      \[x=5/4\]