If `ax^(2) + by^(2) = 1, a_(1) x^(2) + b_(1)y^(2) = 1 ` , then show that the condition for orthogonality of above curves is `(1)/(a)-(1)/(b)=(1)/(a_(1))-(1)/(b_(1))`

Find the condition for the orthogonality of the curves ax^2 + by^2 = 1 and a_1 x^2 + b_1 y^2 = 1

If one of the lines given by the equation a_(1)x^(2)+2h_(1)xy+b_(1)y^(2)=0 coincides with one of the lines given by a_(2)x^(2)+2h_(2)xy+b_(2)y^(2)=0 and the other lines represented by them are perpendecular then prove that. h_(1)((1)/(a_(1))-(1)/(b_(1)))=h_(2)((1)/(a_(2))-(1)/(b_(2)))

If the pair of lines a_(1)x^(2)+2h_(1)xy+b_(1)y^(2)=0 and a_(2)x^(2)+2h_(2)xy+b_(3)y^(2)=0 have one line in common then show that (a_(1)b_(2)-a_(2)b_(1))^(2)+4(a_(1)h_(2)-a_(2)h_(1))(b_(1)h_(2)-b_(2)h_(1))=0

Theorem : A necessary and sufficient condition for the quadratic equations a_(1)x^(2)+b_(1)x+c_(1)=0" and "a_(2)x^(2)+b_(2)x+c_(2)=0 to have a common roots is (c_(1)a_(2)-c_(2)a_(1))^(2)=(a_(1)b_(2)-a_(2)b_(1))(b_(1)c_(2)-b_(2)c_(1))

The Cartesian equation of the sides BC,CA, AB of a triangle are respectively u_(1)=a_(1)x+b_(1)y+c_(1)=0, u_(2)=a_(2)x+b_(2)y+c_(2)=0 and u_(3)=a_(3)x+b_(3)y+c_(3)=0 . Show that the equation of the straight line through A bisectig the side bar(BC) is (u_(3))/(a_(3)b_(1)-a_(1)b_(3))=(u_(2))/(a_(1)b_(2)-a_(2)b_(1))

If (a_(1), b_(1), c_(1)), (a_(2), b_(2), c_(2)) are d.r.'s of two perpendicular lines then

Find the equation of locus of point equidistant from the points (a_(1)b_(1)) and (a_(2),b_(2)) .

If (a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)) are direction ratio of two lines such that (a_(1)^(2)+b_(1)^(2)+c_(1)^(2))(a_(1)a_(2)+b_(1)b_(2)+c_(1)c_(2))^(2)=0 then the two lines