Question

Find the single equation of a pair of straight lines passing through the origin and perpendicular to the lines represented by the equation x2 - xy - 2y2 = 0​

Answer 1

Answer:

$Multiply the equation by bb and then complete the square in yy:</p><p></p><p>(by−hx)2−(h2−ab)x2=0(by−hx)2−(h2−ab)x2=0</p><p></p><p>Solve for yy:</p><p></p><p>y=1b(h±h2−ab−−−−−−√)xy=1b(h±h2−ab)x</p><p></p><p>So these are the two lines through the origin represented by the given equation. The perpendicular lines to these are:</p><p></p><p>x=1b(−h∓h2−ab−−−−−−√)yx=1b(−h∓h2−ab)y</p><p></p><p>In other words the roles of xx and yy have swapped and the sign of hh has changed. Therefore the single equation of perpendicular straight lines is:</p><p></p><p>bx2+2hxy+ay2=0</p><p></p><p></p><p>$

Explanation:

Multiply the equation by b and then complete the square in y :

(by−hx)2−(h2−ab)x2=0

Solve for y :

y=1b(h±h2−ab−−−−−−√)x

So these are the two lines through the origin represented by the given equation. The perpendicular lines to these are:

x=1b(−h∓h2−ab−−−−−−√)y

In other words the roles of x and y have swapped and the sign of h has changed. Therefore the single equation of perpendicular straight lines is:

bx2+2hxy+ay2=0