Question

If the lines \( x + (k - 1)y + 3 = 0 \) \& \( 2x + k^2y - 4 = 0 \) are perpendicular and their point of intersection is the centre of a circle which passes through origin. If chord \( x - y + 2 = 0 \) intersects this circle at \( A \) \& \( B \) then \( (AB)^2 = ? \)

• A. 18
• B. 20
• C. 9
• D. 36

Hint:

If a chord is at perpendicular distance \( d \) from the centre of a circle of radius \( r \), then its length is \( 2\sqrt{r^2-d^2} \). Squaring often makes the calculation faster.

Answer 1

The Correct Option is A

Step 1: Use the condition that the given lines are perpendicular.
The slopes of the lines are
\[ m_1 = -\frac{1}{k-1}, \qquad m_2 = -\frac{2}{k^2}. \] Since the lines are perpendicular,
\[ m_1m_2 = -1. \] So,
\[ \left(-\frac{1}{k-1}\right)\left(-\frac{2}{k^2}\right) = -1. \] Hence,
\[ \frac{2}{k^2(k-1)} = -1. \] Therefore,
\[ k^2(k-1) = -2 \] \[ k^3 - k^2 + 2 = 0. \] On factorising,
\[ (k+1)(k^2-2k+2)=0. \] Thus, the only real value of \( k \) is
\[ k=-1. \]
Step 2: Find the point of intersection of the two lines.
Substituting \( k=-1 \), the lines become
\[ x-2y+3=0 \] and
\[ 2x+y-4=0. \] From the second equation,
\[ y=4-2x. \] Substitute into the first equation:
\[ x-2(4-2x)+3=0 \] \[ x-8+4x+3=0 \] \[ 5x-5=0 \] \[ x=1. \] Then,
\[ y=4-2(A)=2. \] So, the centre of the circle is
\[ C(1,2). \]
Step 3: Find the radius of the circle.
The circle passes through the origin \( O(0,0) \).
Hence, radius is the distance between \( C(1,2) \) and \( O(0,0) \):
\[ r=\sqrt{(1-0)^2+(2-0)^2}=\sqrt{1+4}=\sqrt{5}. \] So,
\[ r^2=5. \]
Step 4: Find the perpendicular distance from the centre to the chord line.
The chord is given by
\[ x-y+2=0. \] Distance of point \( (1,2) \) from this line is
\[ d=\frac{|1-2+2|}{\sqrt{1^2+(-1)^2}}=\frac{1}{\sqrt{2}}. \] Thus,
\[ d^2=\frac{1}{2}. \]
Step 5: Use the chord length formula.
For a chord at distance \( d \) from the centre of a circle of radius \( r \),
\[ AB = 2\sqrt{r^2-d^2}. \] Therefore,
\[ (AB)^2 = 4(r^2-d^2). \] Substituting the values,
\[ (AB)^2 = 4\left(5-\frac{1}{2}\right) \] \[ =4\left(\frac{9}{2}\right) \] \[ =18. \] Final Answer: \( 18 \)