Question

The condition that the lines joining the origin to the points of intersection of the line \( \frac{x}{a} + \frac{y}{b} = 2 \) and the circle \( (x - a)^2 + (y - b)^2 = r^2 \) are at right angles is:

• A. \( a^2 + b^2 = r^2 \)
• B. \( a^2 - b^2 = r^2 \)
• C. \( a^2 - b^2 + r^2 = 0 \)
• D. \( a^2 + b^2 + r^2 = 0 \)

Hint:

When analyzing geometric conditions involving perpendicular lines, focus on using the product of slopes and derive relations using geometry.

Answer 1

The Correct Option is A

Step 1: Understand the condition for perpendicularity.
For the lines joining the origin to the points of intersection of the line and circle to be perpendicular, the product of their slopes should be \( -1 \). This condition leads to a relationship between the geometry of the line and the circle.
Step 2: Analyze the geometric condition.
The points of intersection are derived by solving the system of the line equation \( \frac{x}{a} + \frac{y}{b} = 2 \) and the circle equation \( (x - a)^2 + (y - b)^2 = r^2 \). By applying the perpendicularity condition and solving, we get the condition: \[ a^2 + b^2 = r^2 \]