The locus of the point of intersection of the lines $(\sqrt{3}) k x+k y-4 \sqrt{3}=0$ and $\sqrt{3} x-y-4(\sqrt{3}) k=0$ is a conic, whose eccentricity is ____

JEE Main · 2021 · 25 Feb 2021 · Shift 1 · Circle

The locus of the point of intersection of the lines

(3) k x + k y - 43 = 0

and

3 x - y - 4 (3) k = 0

is a conic, whose eccentricity is ____

Answer: 2

Solution

Writing the first line as

k (3 x + y) = 43

gives

3 x + y = \frac{4 3}{k}

, and the second gives

3 x - y = 43 k

. Multiplying the two eliminates

k

3 x^{2} - y^{2} = 48, \frac{x ^{2}}{16} - \frac{y ^{2}}{48} = 1.

This is a hyperbola with

a^{2} = 16

and

b^{2} = 48

, so its eccentricity is

1 + \frac{48}{16} = 2

JEE Main · 2021 · 25 Feb 2021 · Shift 1 · Integer answer

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