The locus of the point of intersection of the | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Given lines are:$L_1: \sqrt{2}x - y + 4\sqrt{2}k = 0 \Rightarrow y = \sqrt{2}x + 4\sqrt{2}k \quad (i)$$L_2: \sqrt{2}kx + ky - 4\sqrt{2} = 0 \quad

Vedclass · Accepted Answer

$A$ hyperbola with length of its transverse axis $8\sqrt{2}$

Vedclass · Answer

(A) Given lines are:$L_1: \sqrt{2}x - y + 4\sqrt{2}k = 0 \Rightarrow y = \sqrt{2}x + 4\sqrt{2}k \quad (i)$$L_2: \sqrt{2}kx + ky - 4\sqrt{2} = 0 \quad (ii)$Substituting $y$ from $(i)$ into $(ii)$:$\sqrt{2}kx + k(\sqrt{2}x + 4\sqrt{2}k) - 4\sqrt{2} = 0$$\sqrt{2}kx + \sqrt{2}kx + 4\sqrt{2}k^2 - 4\sqrt{2} = 0$$2\sqrt{2}kx = 4\sqrt{2}(1 - k^2)$$x = \frac{2(1 - k^2)}{k}$Substituting $x$ back into $(i)$:$y = \sqrt{2}(\frac{2(1 - k^2)}{k}) + 4\sqrt{2}k = \frac{2\sqrt{2} - 2\sqrt{2}k^2 + 4\sqrt{2}k^2}{k} = \frac{2\sqrt{2}(1 + k^2)}{k}$From these,we observe the relationship between $x$ and $y$:$\frac{y}{4\sqrt{2}} = \frac{1+k^2}{2k}$ and $\frac{x}{4} = \frac{1-k^2}{2k}$$(\frac{y}{4\sqrt{2}})^2 - (\frac{x}{4})^2 = (\frac{1+k^2}{2k})^2 - (\frac{1-k^2}{2k})^2 = \frac{(1+k^2)^2 - (1-k^2)^2}{4k^2} = \frac{4k^2}{4k^2} = 1$This is the equation of a hyperbola $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$ where $a = 4\sqrt{2}$.The length of the transverse axis is $2a = 2(4\sqrt{2}) = 8\sqrt{2}$.

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

Similar Questions

The locus of the centers of the circles touching the lines $3x - 4y + 1 = 0$ and $12x + 5y - 1 = 0$ is/are:

The locus of the mid-points of the chords of the circle $x^2+y^2=16$ which are the tangents to the hyperbola $9x^2-16y^2=144$ is

$A$ point moves in such a way that its distance from the origin is always $4$. Then the locus of the point is:

The locus of a point $(x, y)$ whose distance from the point $(-g, -f)$ is always $a$,where $k = g^2 + f^2 - a^2$,is:

The locus of a point which moves so that the ratio of the lengths of the tangents to the circles $x^2 + y^2 + 4x + 3 = 0$ and $x^2 + y^2 - 6x + 5 = 0$ is $2:3$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module