Question

Let a circle pass through the origin and its center be the point of intersection of two mutually perpendicular lines \( x + (k-1)y + 3 = 0 \) and \( 2x + k2y - 4 = 0 \). If the line \( x - y + 2 = 0 \) intersects the circle at the points A and B, then \( (AB)^2 \) is equal to:

• A. 10
• B. 27
• C. 18
• D. 34

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Two lines are geometrically perpendicular if the product of their slopes is \(-1\).
After finding \(k\), we can find the intersection of the two lines to determine the center of the circle \(C\).
The distance from the center \(C\) to the origin \((0,0)\) directly gives the radius \(r\).
Using the perpendicular distance \(d\) from \(C\) to the given secant line, we can find the length of the chord \(AB\).
Step 2: Key Formula or Approach:
First, identify the slopes of the given lines.
For \(L_1: x + (k - 1)y + 3 = 0\), the slope is \(m_1 = \frac{-1}{k-1}\).
For \(L_2: 2x + k^2y - 4 = 0\), the slope is \(m_2 = \frac{-2}{k^2}\).
The condition for perpendicular lines is \(m_1 m_2 = -1\).
The length of a chord is given by Pythagoras theorem: \(AB = 2\sqrt{r^2 - d^2}\).
Step 3: Detailed Explanation:
Apply the perpendicularity condition.
\[ \left(\frac{-1}{k-1}\right) \left(\frac{-2}{k^2}\right) = -1 \] \[ \frac{2}{k^2(k-1)} = -1 \implies k^3 - k^2 + 2 = 0 \] By inspection of small integers, \(k = -1\) satisfies the equation since \((-1)^3 - (-1)^2 + 2 = -1 - 1 + 2 = 0\).
Substitute \(k = -1\) back into the equations of the lines to get their concrete forms.
\(L_1: x - 2y + 3 = 0\)
\(L_2: 2x + y - 4 = 0 \implies y = 4 - 2x\)
Substitute \(y\) from \(L_2\) into \(L_1\) to find the intersection.
\[ x - 2(4 - 2x) + 3 = 0 \implies x - 8 + 4x + 3 = 0 \implies 5x = 5 \implies x = 1 \] \[ y = 4 - 2(1) = 2 \] So, the center of the circle is \(C(1, 2)\).
Since the circle passes through the origin \((0,0)\), the radius squared is the squared distance to the origin.
\[ r^2 = (1 - 0)^2 + (2 - 0)^2 = 1 + 4 = 5 \] Now, calculate the perpendicular distance \(d\) from the center \(C(1, 2)\) to the line \(x - y + 2 = 0\).
\[ d = \frac{|1(1) - 1(2) + 2|}{\sqrt{1^2 + (-1)^2}} = \frac{|1|}{\sqrt{2}} = \frac{1}{\sqrt{2}} \] The length of the chord squared can now be computed.
\[ (AB)^2 = (2\sqrt{r^2 - d^2})^2 = 4(r^2 - d^2) \] Substitute the values \(r^2 = 5\) and \(d^2 = \frac{1}{2}\) into the expression.
\[ (AB)^2 = 4\left(5 - \frac{1}{2}\right) = 4\left(\frac{9}{2}\right) = 18 \] Step 4: Final Answer:
The value of \((AB)^2\) is \(18\).