Question

If the chord of contact of the point \( P(h, k) \) with respect to the circle \( x^2 + y^2 - 4x - 4y + 8 = 0 \) meets the circle in two distinct points and it also makes an angle \( 45^\circ \) with the positive X-axis in the positive direction, then \( (h, k) \) cannot be:

• A. \( \left( \frac{5}{3}, \frac{2}{2} \right) \)
• B. \( \left( \frac{5}{7}, \frac{3}{3} \right) \)
• C. \( (3, 1) \)
• D. \( (2, 2) \)

Hint:

When solving for the chord of contact, always use the point-slope form and apply the given angle condition for solving the geometry.

Answer 1

The Correct Option is D

We are given a circle with the equation \( x^2 + y^2 - 4x - 4y + 8 = 0 \). The chord of contact for the point \( P(h, k) \) intersects the circle at two distinct points and forms a \( 45^\circ \) angle with the positive X-axis. The condition for the point \( (h, k) \) is given by the geometry of the situation. The equation of the chord of contact can be written as: \[ h(x - 2) + k(y - 2) = 8 \] Now, applying the condition that the line makes a \( 45^\circ \) angle with the X-axis, we solve for the values of \( h \) and \( k \). After solving, we find that the point \( (2, 2) \) does not satisfy this condition. Therefore, the correct answer is \( (2, 2) \).