p₁ and p₂ are the perpendicular distances from the origin to the tangent and normal drawn at any point on the curve x^ 2 / 3 +y^ 2 / 3 =a^ 2 / 3 respectively. If k₁ p₁^2+k₂ p₂^2=a^2 , then k₁+k₂=

Step-by-step solution

x^ 2 3 +y^ 2 3 =a^ 2 3 .... (i) Any point on (i) is (a ^3 , a ^3 ) Slope of tangent =- Equation of the tangent is : aligned & y-a ^3 =- (x-a ^3 ) & x +y = a 2 2 .... (ii) aligned Slope of normal = Equation of normal is : y-a ^3 = (x-a ^3 ) x -y =a 2 Now, p₁= | -a ^2 + ^2 |= a 2 2 2 p₁=a 2 .... (iii) and p₂= | -a 2 ^2 + ^2 |=a 2 p₂=a 2 Squaring and adding (iii) and (iv) aligned & 4 p₁^2+p₂^2=a^2 & k₁=4, k₂=1 k₁+k₂=5 aligned