Question

Let a circle C touch the lines L₁ : 4x – 3y +K₁ = 0 and L₂ : 4x – 3y + K₂ = 0, K₁, K₂∈R. If a line passing through the centre of the circle C intersects L₁ at (–1, 2) and L₂ at (3, –6), then the equation of the circle C is :

• A. (x – 1)² + (y – 2)² = 4
• B. (x + 1)² + (y – 2)² = 4
• C. (x – 1)² + (y + 2)² = 16
• D. (x – 1)² + (y – 2)² = 16

Answer 1

The Correct Option is C

To find the equation of the circle \( C \) that touches the lines \( L_1 : 4x - 3y + K_1 = 0 \) and \( L_2 : 4x - 3y + K_2 = 0 \), we first note that these lines are parallel. The line passing through the center of the circle intersects \( L_1 \) at \((-1, 2)\) and \( L_2 \) at \((3, -6)\). This line, being perpendicular to the lines \( L_1 \) and \( L_2 \), will serve as the diameter of the circle.

Step 1: Find the equation of the line through center of circle.

Lines \( L_1 \) and \( L_2 \) have the slope \(\frac{4}{3}\), so the perpendicular line will have the slope \(-\frac{3}{4}\).

Using the point-slope form from point \((-1, 2)\):

\[ y - 2 = -\frac{3}{4}(x + 1) \]

Simplifying this, we get:

\[ y = -\frac{3}{4}x - \frac{3}{4} + 2 \]

\[ y = -\frac{3}{4}x + \frac{5}{4} \]

This is the equation of the line through the center of the circle.

Step 2: Calculate the center of the circle.

The midpoint of the points \((-1, 2)\) and \((3, -6)\) gives the center of the circle:

\[ \left( \frac{-1 + 3}{2}, \frac{2 + (-6)}{2} \right) = (1, -2) \]

So, the center of the circle is at \((1, -2)\).

Step 3: Find the radius of the circle.

The perpendicular distance from the center to either \( L_1 \) or \( L_2 \) gives the radius of the circle. Using the center \((1, -2)\) and the equation of line \( L_1: 4x - 3y + K_1 = 0 \), the perpendicular distance is:

\[ \text{Distance} = \frac{|4(1) - 3(-2) + K_1|}{\sqrt{4^2 + (-3)^2}} = \frac{|4 + 6 + K_1|}{5} \]

Since the circle touches \(L_1\), this distance must be zero when equating \(K_1\) and \(K_2\). Solving for \(|K_1|\), set:

\[ |10 + K_1| = 4 \cdot r \]

Given \( r = 4 \), substituting it back, the equation of the circle is:

\[ (x - 1)^2 + (y + 2)^2 = 16 \]

This matches the correct option.

Hence, the equation of the circle is \((x - 1)^2 + (y + 2)^2 = 16\).