Straight Line Question 152

Question: If the quadrilateral formed by the lines $ ax+by+c=0,a’x+b’y+c’=0,ax+by+c’=0,a’x+b’y+c’=0 $ has perpendicular diagonals, then

Options:

A) $ b^{2}+c^{2}=b{{’}^{2}}+c{{’}^{2}} $

B) $ c^{2}+a^{2}=c{{’}^{2}}+a{{’}^{2}} $

C) $ a^{2}+b^{2}=a{{’}^{2}}+b{{’}^{2}} $

D) none of these

Show Answer

Answer:

Correct Answer: C

Solution:

  • [c] Since the diagonals are perpendicular, the given quadrilateral is a rhombus. So the distances between two pairs of parallel sides are equal, hence, $ | \frac{c’-c}{\sqrt{a^{2}+b^{2}}} |=| \frac{c’-c}{\sqrt{a{{’}^{2}}+b{{’}^{2}}}} | $ Or $ a^{2}+b^{2}=a{{’}^{2}}+b{{’}^{2}} $