Let A = (a, 0) and B = (-a, 0) be two fixed p | vedclass

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

Question

(B) Let $P = (x, y)$. The condition $PA = nPB$ implies $PA^2 = n^2 PB^2$.$(x - a)^2 + y^2 = n^2((x + a)^2 + y^2)$.$x^2 - 2ax + a^2 + y^2 = n^2(x^2 + 2

Vedclass · Accepted Answer

$A$ lies outside the circle and $B$ lies inside the circle.

Vedclass · Answer

(B) Let $P = (x, y)$. The condition $PA = nPB$ implies $PA^2 = n^2 PB^2$.$(x - a)^2 + y^2 = n^2((x + a)^2 + y^2)$.$x^2 - 2ax + a^2 + y^2 = n^2(x^2 + 2ax + a^2 + y^2)$.$(n^2 - 1)x^2 + (n^2 - 1)y^2 + 2ax(n^2 + 1) + a^2(n^2 - 1) = 0$.Dividing by $(n^2 - 1)$,we get $x^2 + y^2 + 2ax \frac{n^2 + 1}{n^2 - 1} + a^2 = 0$.This is the equation of a circle with center $C = (-a \frac{n^2 + 1}{n^2 - 1}, 0)$ and radius $r = |\frac{2an}{n^2 - 1}|$.Since $a  0$. Then the center is $C = (k \frac{n^2 + 1}{n^2 - 1}, 0)$.For $n > 1$,the circle encloses the point $B(-a, 0)$ because $B$ is closer to the center relative to the radius,while $A(a, 0)$ lies outside.

Let $A = (a, 0)$ and $B = (-a, 0)$ be two fixed points. For $a \in (-\infty, 0)$,point $P$ moves in the plane such that $PA = nPB$ $(n \neq 0, n \neq 1)$. If $n > 1$,then which of the following is true regarding the positions of $A$ and $B$ with respect to the locus of $P$?

Similar Questions

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:

If the points $A(2,3)$ and $B(3,2)$ form a triangle with a variable point $P(t, t^2)$,where $t$ is a parameter,then the equation of the locus of the centroid of triangle $ABP$ is:

What does the equation $\sqrt{(x - 2)^2 + (y + 1)^2} + \sqrt{(x + 2)^2 + (y + 4)^2} = 5$ represent?

$A$ point moves such that the sum of the squares of its distances from the points $(1, 2)$ and $(-2, 1)$ is always $6$. Then, its locus is

The locus of the centers of the circles that are passing through the intersection of the circles $x^2+y^2=1$ and $x^2+y^2-2x+y=0$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module