 Hi and welcome to the session. Let us discuss the following question. Question says, prove that if a plane has the intercepts a, b, c and is it a distance of p units from the origin, then 1 upon a square plus 1 upon b square plus 1 upon c square is equal to 1 upon p square. First of all, let us understand that if equation of the plane is ax plus by plus cz plus d is equal to 0, then intercepts a, b, c can be written as a is equal to minus d upon a, b is equal to minus d upon b and c is equal to minus d upon b. This is the key idea to solve the given question. Let us now start with the solution. Now we are given plane has intercepts a, b, c. Now let us assume that plane is of the form ax plus by plus cz is equal to minus d. Now we are given in the question that d is equal to p. So we can write this equation as ax plus by plus cz is equal to minus p. We know d is the distance of the plane from the origin. Here we are given that distance of the plane from the origin is p. Now from key idea we know intercept a is equal to minus p upon a. Now this further implies a is equal to minus p upon a. Also intercept b is equal to minus p upon b. Now this further implies b is equal to minus p upon intercept b. Similarly c is equal to minus p upon c. This implies c is equal to minus p upon intercept c. Now we know a, b and c are direction cosines of the normal to the plane. So a square plus b square plus c square is equal to 1. Now let us name this expression as 1. Now we will substitute values of a, b and c in this expression. Now we get square of minus b upon a plus square of minus b upon b plus square of minus b upon c is equal to 1. Now this further implies p square upon a square plus p square upon b square plus p square upon c square is equal to 1. Now multiplying both the sides of this equation by 1 upon p square we get 1 upon a square plus 1 upon b square plus 1 upon c square is equal to 1 upon p square. So we get if a plane has intercepts a, b, c and is at a distance of p units from the origin then 1 upon a square plus 1 upon b square plus 1 upon c square is equal to 1 upon p square. So this is our required proof. This completes the session. Hope you understood the solution. Take care and have a nice day.