Question

If the equation \(x^{2}-y^{2}-x+\lambda y-2=0\) (where \(\lambda >0\)) represents a pair of lines, then the coordinates of their point of intersection are:

• A. \((1,3)\)
• B. \(\left(\frac{1}{2},\frac{3}{2}\right)\)
• C. \(\left(\frac{3}{2},\frac{1}{2}\right)\)
• D. \((0,-1)\)

Hint:

Use the determinant condition for a pair of lines to find lambda, then solve the partial derivatives.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
Compare with \(ax^2+2hxy+by^2+2gx+2fy+c=0\): \(a=1,b=-1,h=0,g=-\dfrac{1}{2},f=\dfrac{\lambda}{2},c=-2\).

Step 2: Condition for a Pair of Lines
\(abc+2fgh-af^2-bg^2-ch^2=0\)
\(2-\dfrac{\lambda^2}{4}+\dfrac{1}{4}=0\Rightarrow\lambda^2=9\).

Step 3: Find lambda
Since \(\lambda>0\), \(\lambda=3\).

Step 4: Intersection Point
Partial derivatives:
\(2x-1=0\Rightarrow x=\dfrac{1}{2}\)
\(-2y+3=0\Rightarrow y=\dfrac{3}{2}\).

Final Answer:
The point of intersection is \(\left(\dfrac{1}{2},\dfrac{3}{2}\right)\). \[ \boxed{\text{(B) }\left(\dfrac{1}{2},\dfrac{3}{2}\right)} \]