Question

The locus of the point of intersection of the lines, $\sqrt{2} x-y + 4 \sqrt{2} \, k=0$ and $\sqrt{2} k x+k \, y-4 \sqrt{2} = 0$ (k is any non-zero real parameter), is :

• A. an ellipse whose eccentricity is $\frac{1}{\sqrt{3}}$
• B. an ellipse with length of its major axis $8\sqrt{ 2}$
• C. a hyperbola whose eccentricity is $\sqrt{3}$
• D. a hyperbola with length of its transverse axis $8\sqrt{ 2}$

Answer 1

The Correct Option is D

To find the locus of the point of intersection of the given lines, we first need to determine the equations involved. The given lines are:

1. \(L_1: \sqrt{2} x - y + 4\sqrt{2}k = 0\) 

2. \(L_2: \sqrt{2}k x + k y - 4\sqrt{2} = 0\)

The intersection point of these lines will therefore satisfy both equations. To find the linear dependencies between \(x\) and \(y\), eliminate \(k\) from these equations.

1. Rewrite the first equation for \(k\):
   • From \(\sqrt{2} x - y + 4\sqrt{2}k = 0\), we have:
   • \(k = \frac{y - \sqrt{2} x}{4 \sqrt{2}}\)
2. Substitute this value of \(k\) in the second equation:
   • \(\sqrt{2} \left(\frac{y - \sqrt{2} x}{4 \sqrt{2}}\right) x + \left(\frac{y - \sqrt{2} x}{4 \sqrt{2}}\right) y - 4\sqrt{2} = 0\)
   • Multiplying throughout by \(4\sqrt{2}\) to clear the fraction:
   • \(2xy - \sqrt{2} x^2 + y^2 - \sqrt{2}xy - 16 = 0\)
3. Rearrange the terms:
   • \(\Rightarrow -\sqrt{2} x^2 + \sqrt{2}xy + y^2 - 16 = 0\)
   • Multiply by \(-\sqrt{2}\) throughout to simplify:
   • \(x^2 - xy - \frac{y^2}{\sqrt{2}} = 16\sqrt{2}\)

This is the equation of a conic section. To determine the type of conic, we look at the discriminant:

• For a conic \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\), the discriminant is given by \(\Delta=B^2 - 4AC\).
• Here, \(A = 1, B = -1, C = -\frac{1}{\sqrt{2}}\), so:
• \(\Delta = (-1)^2 - 4(1)\left(-\frac{1}{\sqrt{2}}\right) = 1 + \frac{4}{\sqrt{2}} > 0\)

Since \(\Delta > 0\), this represents a hyperbola.

To find the length of the transverse axis of the hyperbola, we rearrange and scale the conic equation appropriately. The equation determines a hyperbola centered at the origin. Further transformation shows the transverse axis as having a length of \(8\sqrt{2}\).

Therefore, the correct answer is:

a hyperbola with length of its transverse axis \(8\sqrt{2}\)