The product of perpendiculars let fall from the point `(x_(1),y_(1))` upon the lines represented by `ax^(2)+2hxy+by^(2)`, is

Options

A. `(|ax_(1)^(2)+2hx_(1)y_(1)+by_(1)^(2)|)/(sqrt((a-b)^(2)+4h^(2)))`
B. `(|ax_(1)^(2)+2hx_(1)y_(1)+by_(1)^(2)|)/(sqrt((a-b)^(2)+h^(2)))`
C. `(|ax_(1)^(2)+2hx_(1)y_(1)+by_(1)^(2)|)/(sqrt((a+b)^(2)+4h^(2)))`
D. `(|ax_(1)^(2)-2hxy_(1)y_(1)+by_(1)^(2)|)/(sqrt((a-b)^(2)+4h^(2)))`

Answer

A

Explanation

Let `y-x_(1)x=0` and `y=m_(2)x=0` be the lines represented by the equation `ax^(2)+2hxy+by^(2)=0`. Then,
`m_(1)+m_(2)=-(2h)/(b) and m_(1)m_(2)=(a)/(b)" (i)"`
Let `p_(1) and p_(2)` be the lengths of perpendiculars from `(x_(1),y_(1))` on lines `y-m_(1)x=0` and `y-m_(2)x=0`. Then,
`p_(1)=(y_(1)-m_(1)x_(1))/(sqrt(1+m_(1)^(2)))andp_(2)=(y_(1)-m_(2)x_(1))/(sqrt(1+x_(2)^(2)))`
`therefore" "p_(1)p_(2)=((y_(1)-m_(1)x_(1))/(sqrt(1+x_(1)^(2))))((y_(1)-m_(2)x_(1))/(sqrt(1+m_(2)^(2))))=((y_(1)-m_(1)x_(1))(y_(1)-m_(2)x_(1)))/(sqrt((1+m_(1)^(2))(1+m_(2))))`
`rArr" "p_(1)p_(2)=(y_(1)^(2)-x_(1)y_(1)(m_(1)+m_(2))+m_(1)m_(2)x_(1)^(2))/(sqrt(1+(m_(1)^(2)+m_(2)^(2))+m_(1)^(2)m_(2)^(2)))`
`rArr" "p_(1)p_(2)=(y_(1)^(2)-x_(1)y_(1)(m_(1)+m_(2))+m_(1)m_(2)x_(1)^(2))/(sqrt(1+(m_(1)+m_(2))^(2)-2m_(1)m_(2)+(m_(1)m_(2))^(2)))`
`rArr" "p_(1)p_(2)=(y_(1)^(2)-x_(1)y_(1)(-(2h)/(b))+(a)/(b)x_(1)^(2))/(sqrt(1+(4h^(2))/(b^(2))-(2a)/(b)+(a^(2))/(b^(2))))" [Using (i)]"`
`rArr" "p_(1)p_(2)=(ax_(1)^(2)+2hx_(1)y_(1)+by_(1)^(2))/(sqrt((a-b)^(2)+4h^(2)))`