Conic Sections Question 141
Question: A set of parallel chords of the parabola $ y^{2}=4ax $ have their mid-point on
Options:
A) Any straight line through the vertex
B) Any straight line through the focus
C) Any straight line parallel to the axis
D) Another parabola
Show Answer
Answer:
Correct Answer: C
Solution:
Let $ y=mx+c $ is chord and c is variable
$ \Rightarrow x=( \frac{y-c}{m} ) $ by $ y^{2}=4ax $
For getting points of intersection, $ y^{2}=4a( \frac{y-c}{m} )\Rightarrow y^{2}-\frac{4ay}{m}+\frac{4ac}{m}=0 $
therefore $ y_1+y_2=\frac{4a}{m}\Rightarrow \frac{y_1+y_2}{2}=\frac{2a}{m} $
which is a constant; independent to c.
BETA
