Question

6. If the distance from P to the points (5,-4), (7,0)
are in the ratio 2:3, then the locus of P is
1) 5x2 + 5y2 - 12x - 86y + 17 = 0
2) 5x2 + 5y2 - 34x + 120y + 29 = 0
3) 5x2 + 5y2 - 5x + y + 14 = 0
4) 3x2 + 3y2 - 20x +38y + 87 = 0
n​

Answer 1

Answer:

B

5x

2

+5y

2

−34x+120y+29=0

Let P(x,y) be a point in locus.

The given points are A(5,−4) and B(7,6) and given that PA:PB=2:3. Therefore, we have:

PB

PA

=

3

2

⇒3PA=2PB

⇒9PA

2

=4PB

2

Now, applying the distance formula to the given points A(5,−4) and B(7,6):

9[(x−5)

2

+(y+4)

2

]=4[(x−7)

2

+(y−6)

2

]

⇒9[x

2

+5

2

−(2×x×5)+y

2

+4

2

+(2×y×4)]=4[x

2

+7

2

−(2×x×7)+y

2

+6

2

−(2×y×6)]

(∵(a+b)

2

=a

2

+b

2

+2ab,(a−b)

2

=a

2

+b

2

−2ab)

⇒9(x

2

+25−10x+y

2

+16+8y)=4(x

2

+49−14x+y

2

+36−12y)

⇒9(x

2

+y

2

−10x+8y+41)=4(x

2

+y

2

−14x−12y+85)

⇒9x

2

+9y

2

−90x+72y+369=4x

2

+4y

2

−56x−48y+340

⇒9x

2

−4x

2

+9y

2

−4y

2

−90x+56x+72y+48y+369−340=0

⇒5x

2

+5y

2

−34x+120y+29=0

Hence, the equation of locus of P is 5x

2

+5y

2

−34x+120y+29=0.