Find the equation of the plane which contains the line of intersection of the planes →r.(^i−^j+3^k)=0 and →r.(−^i+^j+^k)+5=0 and whose intercept on x-axis is equal to that of on y-axis.

Planes are →r (^i−^j+3^k)=0 →r (−^i+^j+^k)+5=0 ⇒x−y+3z=0 −x+y+z+5=0
Any plane passing through the line of intersect. x−y+3z+λ(−x+y+z+5)=0
x(1−λ)+y(−1+λ)+z(3+λ)+(5λ)=0
Intercepts are equal on axes
So, 5λ/(1−λ) = 5λ/(−1+λ) ⇒ −1+λ=1−λ ⇒ 2λ=2 ⇒ λ=1
Therefore, the required plane is −x+4z+5=0 or x−4z−5=0.