Question

The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line \[ 4x - 5y = 20 \] to the circle \[ x^2 + y^2 = 9 \text{ is:} \]

• A. \( 20(x^2 + y^2) - 36x + 45y = 0 \)
• B. \( 20(x^2 + y^2) + 36x - 45y = 0 \)
• C. \( 36(x^2 + y^2) - 20x + 45y = 0 \)
• D. \( 36(x^2 + y^2) + 20x - 45y = 0 \)

Hint:

When solving for the locus of the mid-point of the chord of contact, use the formula for the chord of contact and relate it to the given geometric conditions (such as the equation of the straight line and the circle).

Answer 1

The Correct Option is A

We are given the straight line: \[ 4x - 5y = 20 \] and the equation of the circle: \[ x^2 + y^2 = 9 \] The chord of contact of tangents drawn from any point \( (x_1, y_1) \) on the straight line to the circle is given by the equation: \[ T = 0 \] where \( T \) is the equation of the tangent to the circle. The equation of the tangent to the circle \( x^2 + y^2 = 9 \) at \( (x_1, y_1) \) is: \[ xx_1 + yy_1 = 9 \] Now, we want the locus of the mid-point of the chord of contact, i.e., the mid-point of the tangents from points on the line \( 4x - 5y = 20 \). For the point \( (x_1, y_1) \) lying on the line, we substitute \( x_1 = x \) and \( y_1 = y \) into the equation of the line: \[ 4x - 5y = 20 \] Now, using the fact that the mid-point of the chord of contact is given by the formula: \[ \frac{x_1 + x_2}{2} = x \quad {and} \quad \frac{y_1 + y_2}{2} = y \] After solving the resulting equations and simplifying, we find that the equation of the locus of the mid-point is: \[ 20(x^2 + y^2) - 36x + 45y = 0 \] Thus, the correct answer is Option A.