Let A- -3-0 and B- 3-0 be two fixed points and P moves on a plane such that PA-n- PBn-0- If n 1- then locus of a point P is

The correct option is B a circleGiven, A=( 3,0), B=(3,0) and #160; PA=nPB Let P be (h,k) PA=(h+3,k) PB=(h 3,k) So, PA=nPB ...

Let A≡ -3,0...

Question

Let $A \equiv (- 3, 0)$ and $B \equiv (3, 0)$ be two fixed points and $P$ moves on a plane such that $P A = n \times P B (n > 0)$ . If $n \neq 1$ , then locus of a point $P$ is

a straight line

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a circle

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a parabola

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an ellipse

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Solution

The correct option is B a circle
Given, $A = (- 3, 0), B = (3, 0)$ and $P A = n P B$

Let $P$ be $(h, k)$

$P A = (h + 3, k)$
$P B = (h - 3, k)$
So, $P A = n P B$
$\Rightarrow$ ${(h + 3)}^{2} + k^{2} = n^{2} [{(h - 3)}^{2} + k^{2}]$

$∴$ locus of $P (h, k)$ is
$\Rightarrow$ $x^{2} + 6 x + 9 + y^{2} = n^{2} [x^{2} - 6 x + 9 + y^{2}]$
$\Rightarrow$ $x^{2} (1 - n^{2}) + y^{2} (1 - n^{2}) + 6 x (1 + n^{2}) + 9 (1 - n^{2}) = 0$
$\Rightarrow$ $x^{2} + y^{2} + 6 \frac{(1 + n^{2})}{1 - n^{2}} x + 9 = 0 (∵ n \neq 1)$
$∴$ locus is a circle

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Q. A point moves in a plane so that its distances PA and PB from two fixed points A and B in the plane satisfy the relation PA − PB = k (k ≠ 0), then the locus of P is
(a) a hyperbola
(b) a branch of the hyperbola
(c) a parabola
(d) an ellipse

$A point moves in a plane so that its distances PA and PB from two fixed points A and B in the plane satisfy the relation PA-PB = k (k \neq 0), t h e n t h e l o c u s o f P i s$

Let A≡(−3,0) and B≡(3,0) be two fixed points and P moves on a plane such that PA=n×PB(n>0). If n≠1, then locus of a point P is

The correct option is B a circleGiven, A=(−3,0),B=(3,0) and PA=nPBLet P be (h,k)PA=(h+3,k)PB=(h−3,k)So, PA=nPB⇒(h+3)2+k2=n2[(h−3)2+k2]∴ locus of P(h,k) is⇒ x2+6x+9+y2=n2[x2−6x+9+y2]⇒x2(1−n2)+y2(1−n2)+6x(1+n2)+9(1−n2)=0⇒x2+y2+6(1+n2)1−n2x+9=0(∵n≠1)∴ locus is a circle

Let $A \equiv (- 3, 0)$ and $B \equiv (3, 0)$ be two fixed points and $P$ moves on a plane such that $P A = n \times P B (n > 0)$ . If $n \neq 1$ , then locus of a point $P$ is