If the quadrilateral formed by the lines `a x+b y+c=0,a^(prime)x+b^(prime)y+c=0,a x+b y+c^(prime)=0,a^(prime)x+b^(prime)y+c^(prime)=0` has perpendicular diagonals, then `b^2+c^2=b^('2)+c^('2)` `c^2+a^2=c^('2)+a^('2)` `a^2+b^2=a^('2)+b^('2)` (d) none of these

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

Answer

C

Explanation

Since the diagonal are perpendicular, the given quadrilteral is a rhombus. So, the distances between two pairs of parallel sides are equal. Hence,
`|(c'-c)/(sqrt(a^(2)+b^(2)))| = |(c'-c)/(sqrt(a'^(2)+b'^(2)))|`
`"or " a^(2) +b^(2) = a'^(2) + b'^(2)`