1-2ab-(a^(2)+b^(2))

(i ) If the length of perpendicular from origin to the line ax+by+a+b=0 is p , then show that : p^(2)-1=(2ab)/(a^(2)+b^(2)) (ii) If the length of perpendicular from point (1,1) to the line ax-by+c=0 is unity then show that : (1)/(a)-(1)/(b)+(1)/(C )=(c )/(2ab)

Show that the tangent of an angle between the lines (x)/(a)+(y)/(b)=1 and (x)/(a)-(y)/(b)=1 and (2ab)/(a^(2)-b^(2)) .

Show that the tangent of an angle between the lines (x)/(a)+(y)/(b)=1 and (x)/(a)-(y)/(b)=1 and (2ab)/(a^(2)-b^(2)) .

Solve for x and y: ax+by=1" and "bx+ay=(2ab)/(a^(2)+b^(2)) .

Solve the following simultaneous linear equations in two linear variables by the method of cross-multiplication : ax+by=1, bx+ay=(2ab)/(a^(2)+b^(2))

If ax+by=1 and bx+ay=(2ab)/(a^(2)+b^(2)) then (x^(2)+y^(2))(a^(2)+b^(2)) is equal to

Answer any three questions Using properties of determinants, prove the following abs{:(1+a^2 - b^2,2ab,-2b),(2ab,1-a^(2) +b^(2) ,2a),(2b,-2a,1-a^2 -b^2):}=(1+a^2 +b^2)^3.

By using properties of determinants , show that : {:[( 1+a^(2) -b^(2) ,2ab , -2b),( 2ab, 1-a^(2) +b^(2) , 2a),( 2b, -2a, 1-a^(2) -b^(2)) ]:}=( 1+a^(2) +b^(2)) ^(3)

Solve the following system of equations in x and yax+by=1bx+ay=((a+b)^(2))/(a^(2)+b^(2))-1 or,bx+ay=(2ab)/(a^(2)+b^(2))

a) Prove that int_(a)^(b)f(x)dx=int_(a)^(b)f(a+b-x)dx" and evaluate "int_(pi//6)^(pi//3)(dx)/(1+sqrt(tanx)) b) Prove that |{:(1+a^(2)-b^(2), 2ab, -2b), (2ab, 1-a^(2)+b^(2), 2a), (2, -2a, 1-a^(2)-b^(2)):}|=(1+a^(2)+b^(2))^(3)