Question

If the line \( 4x - 3y + p = 0 \) (where \( p + 3>0 \)) touches the circle \( x^2 + y^2 - 4x + 6y + 4 = 0 \) at the point \( (h,k) \), then the value of \( h - 2k \) is:

• A. \( \frac{-8}{5} \)
• B. \( 2 \)
• C. \( \frac{6}{5} \)
• D. \( 3 \)

Hint:

For tangents to circles, use the perpendicular distance condition and solve for unknowns systematically.

Answer 1

The Correct Option is B

Step 1: Finding the center and radius of the circle.
Rewriting the given circle equation: \[ x^2 + y^2 - 4x + 6y + 4 = 0 \] Complete the square: \[ (x - 2)^2 - 4 + (y + 3)^2 - 9 + 4 = 0 \] \[ (x - 2)^2 + (y + 3)^2 = 9 \] So the center is \( (2,-3) \) and radius is \( r = 3 \). Step 2: Finding the point of tangency.
For a line \( Ax + By + C = 0 \) to be tangent to a circle, the perpendicular distance from the center to the line must be equal to the radius: \[ \frac{|4(2) - 3(-3) + p|}{\sqrt{4^2 + (-3)^2}} = 3 \] \[ \frac{|8 + 9 + p|}{5} = 3 \] Solving for \( p \): \[ |17 + p| = 15 \] \[ p = -2 \quad \text{or} \quad p = -32 \] Step 3: Computing \( h - 2k \).
The point of tangency is obtained by solving: \[ 4h - 3k + p = 0 \] Substituting \( p = -2 \), solving for \( h, k \), we find: \[ h - 2k = 2 \] Thus, the required value is: \[ \mathbf{2} \]