Conic Sections Question 124
Question: A straight line has its extremities on two fixed straight lines and cuts off from them a triangle of constant area $ c^{2} $ .Then the locus of the middle point of the line is
Options:
A) $ 2xy=c^{2} $
B) $ xy+c^{2}=0 $
C) $ 4x^{2}y^{2}=c $
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
[a] Let the given straight line be the axis of coordinates and let the equation of the variable line be $ \frac{x}{a}+\frac{y}{b}=1 $ This line cuts the coordinate axes at A(a, 0) and B(0, b). therefore, Area of $ \Delta AOB=\frac{1}{2}ab=c^{2} $ Or $ ab=2c^{2} $
(i) If (h, k) are the coordinates of the middle point of AB, then $ h=\frac{a}{2} $ and $ k=\frac{b}{2} $ On eliminating a and b form (i) and (ii), we get $ 2hk=c^{2} $
Hence, the locus of (h, k) is $ 2xy=c^{2} $ .
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