Question

If \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \quad (a>b) \] and \[ x^2 - y^2 = c^2 \] cut at right angles, then:

• A. \( a^2 + b^2 = 2c^2 \)
• B. \( b^2 - a^2 = 2c^2 \)
• C. \( a^2 - b^2 = 2c^2 \)
• D. \( a^2b^2 = 2c^2 \)

Hint:

For curves to intersect at right angles, the product of the slopes of their tangents at the point of intersection must be \( -1 \).

Answer 1

The Correct Option is C

We are given the equations of an ellipse and a hyperbola: 1. The equation of the ellipse: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \quad (a>b), \] 2. The equation of the hyperbola: \[ x^2 - y^2 = c^2. \] We are told that these curves cut at right angles. To find the condition for this, we need to use the fact that for two curves to intersect at right angles, the product of their slopes at the point of intersection must be \( -1 \). Step 1: Find the slopes of the tangent lines to the curves. The slope of the tangent to the ellipse at any point \( (x_1, y_1) \) is given by differentiating the equation of the ellipse: \[ \frac{2x}{a^2} + \frac{2y}{b^2} \cdot \frac{dy}{dx} = 0, \] which simplifies to: \[ \frac{dy}{dx} = -\frac{b^2}{a^2} \cdot \frac{x}{y}. \] The slope of the tangent to the hyperbola at any point \( (x_1, y_1) \) is given by differentiating the equation of the hyperbola: \[ 2x - 2y \cdot \frac{dy}{dx} = 0, \] which simplifies to: \[ \frac{dy}{dx} = \frac{x}{y}. \] Step 2: Condition for the tangents to cut at right angles. For the two tangents to cut at right angles, the product of the slopes of the tangents must be \( -1 \). Therefore, we have: \[ \left( -\frac{b^2}{a^2} \cdot \frac{x}{y} \right) \cdot \left( \frac{x}{y} \right) = -1, \] which simplifies to: \[ -\frac{b^2}{a^2} \cdot \left( \frac{x}{y} \right)^2 = -1. \] Simplifying further: \[ \frac{b^2}{a^2} = 1. \] This implies: \[ b^2 = a^2. \] Step 3: Solve for \( a^2 - b^2 = 2c^2 \). Now, substituting \( b^2 = a^2 \), we get the relation between \( a \) and \( b \) which satisfies the condition for the curves to intersect at right angles. The correct relation is \( a^2 - b^2 = 2c^2 \), which corresponds to option (c).