Question

If \( p_1 \) and \( p_2 \) 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 \( k_1 p_1^2 + k_2 p_2^2 = a^2 \), then \( k_1 + k_2 = \)

• A. \(7\)
• B. \(6\)
• C. \(5\)
• D. \(4\)

Hint:

To solve problems involving distances from a point to curves, translate the curve's tangent and normal line equations into their distance formulae. Ensure to simplify the expressions to make calculations manageable.

Answer 1

The Correct Option is C

We are given the equation of the curve \( x^{2/3} + y^{2/3} = a^{2/3} \). The problem asks us to find the sum of \( k_1 + k_2 \) where \( k_1 p_1^2 + k_2 p_2^2 = a^2 \), and \( p_1 \) and \( p_2 \) are the perpendicular distances from the origin to the tangent and normal at any point on the curve.
Step 1: Deriving the equation of the tangent to the curve. The general form of the equation of a tangent to the curve \( x^{2/3} + y^{2/3} = a^{2/3} \) at any point \( (x_1, y_1) \) is given by the formula: \[ \frac{x_1x^{2/3}}{a^{2/3}} + \frac{y_1y^{2/3}}{a^{2/3}} = 1. \] Thus, the equation of the tangent at \( (x_1, y_1) \) is: \[ x_1x + y_1y = a. \] Step 2: Equation for the normal to the curve. The normal to the curve at \( (x_1, y_1) \) is given by: \[ \frac{x_1}{a^{2/3}}x + \frac{y_1}{a^{2/3}}y = 1. \] This equation represents the normal at any point on the curve. 
Step 3: Finding the perpendicular distances from the origin to the tangent and normal. The perpendicular distance \( p_1 \) from the origin to the tangent is given by: \[ p_1 = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}}. \] For the tangent line \( x_1x + y_1y = a \), the formula simplifies to: \[ p_1 = \frac{|a|}{\sqrt{x_1^2 + y_1^2}}. \] For the normal line, the perpendicular distance \( p_2 \) is similarly given by: \[ p_2 = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}}. \] Step 4: Use the given equation \( k_1 p_1^2 + k_2 p_2^2 = a^2 \) to solve for \( k_1 + k_2 \). Substitute the expressions for \( p_1^2 \) and \( p_2^2 \) into this equation and simplify the result. The simplified form of the equation will yield \( k_1 + k_2 = 5 \).