Single Choice

The locus of the point of intersection of the lines $$\sqrt{3}x - y - 4\sqrt{3}t = 0$$ and $$\sqrt{3}tx + ty - 4\sqrt{3} = 0$$ ( where t is a parameter) is a hyperbola whose eccentricity is

A$$\sqrt{3}$$
B2
Correct Answer
C$$\dfrac{2}{\sqrt{3}}$$
D$$\dfrac{4}{3}$$

Solution

Given, $$\sqrt{3}x - y - 4\sqrt{3}t = 0$$ and $$\sqrt{3}tx + ty - 4\sqrt{3} = 0$$
Eliminating $$t$$ from the given two equation,
from first equation we have,
$$t=\dfrac{\sqrt{3}x - y}{4\sqrt3}$$
And from second we have,
$$t=\dfrac{4\sqrt3}{\sqrt3 x+y}$$
we have $$ \dfrac{x^{2}}{16}-\dfrac{y^{2}}{48}=1 $$
whose eccentricity is $$ e=\sqrt{1+\dfrac{b^2}{a^2}}=\sqrt{1+\dfrac{48}{16}}=2 $$

SIMILAR QUESTIONS

Hyperbola

\The locus of the foot of perpendicular from the focus on any tangent to the hyperbola $$\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1$$ is:

Hyperbola

Let the locus of the middle points of normal chords of the rectangular hyperbola $$x^2-y^2 = a^2$$ be $$(y^2 - x^2)^m = ka^2x^2y^2$$. Find $$k+m$$ ?

Hyperbola

The perpendicular from the centre upon the normal on any point of the hyperbola $$\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$$ meets at $$R$$ . Find the locus of $$R$$.

Hyperbola

Given the base of a triangle and the ratio of the tangent of half the base angles,find the locus of vertex.

Hyperbola

Locus of the feet of the perpendiculars drawn from either focus on a variable tangent to the hyperbola $$16y^{2} - 9x^{2} = 1$$ is

Hyperbola

Locus of a point whose chord of contact with respect to the circle $$x^{2} + y^{2} = 4$$ is a tangent to the hyperbola xy = 1 is a/an

Hyperbola

If tangents PQ and PR are deawn from variable point P to the hyperbola $$\dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = 1 (a> b)$$ so that the fourth vertex S of parallelogram PQSR lies on circumcircle of triangle PQR, then locus of P is

Hyperbola

If the sum of the slopes of normal from a point P to the hyperbola $$xy = c^{2}$$ is equal to $$\lambda (\lambda \epsilon R^{+})$$, then locus of point P is

Hyperbola

If locus of a point, whose chord of contact with respect to the circle $$ x^{2}+y^{2}=4 $$ is a tangent to the hyperbola xy=1 is $$ xy=c^{2} $$ , then value of $$ c^{2} $$ is

Hyperbola

Tangents are drawn from any point on the hyperbola $$ \dfrac{x^{2}}{9}-\dfrac{y^{2}}{4} $$ =1 to the circle $$ x^{2}+y^{2}=9 $$. Find the locus of midpoint of the chord of contact.

Contact Details