Question

If the distance of the point P from the point (3, 4) is √10 and abscissa of P is double its
ordinate, find the co-ordinates of P.

Answer 1

Given:

the distance of the point P from the point (3, 4) is √10

To find:

The co-ordinates of P

Solution:

Let the coordinates of the P is (x,y)

and it is given in the question that

abscissa of P is double its  ordinate

x = 2y

So the coordinate is given by:

(2y, y)

BY the distance formula we get

• $\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}} = \sqrt{10}$
• $\sqrt{(2y-3)^{2}+(y-4)^{2}} = \sqrt{10}$
• $4y^{2}+9-12y+y^{2}+16-8y = 10$
• $5y^{2}-20y+25 = 10$
• $y^{2}-4y+5 = 2$
• $y^{2}-4y+3 = 0$
• $y^{2}-3y-y+3 = 0$
• $y(y-1)-3(y-1) = 0$
• $(y-1)(y-3) = 0$
• $y = 1~or~3$

When y = 1

Point is (2y, y) = (2,1)

When y = 3

Point is (2y, y) = (6,3)

The co-ordinates of P are (2,1) and (6,3)

Answer 2

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

GIVEN

• The distance of the point P from the point (3, 4) is √10

• Abscissa of P is double its ordinate

TO DETERMINE

The co-ordinates of P

FORMULA TO BE IMPLEMENTED

$\sf{The \: distance \: between \: two \: points \: (x_1,y_1) \: and \: (x_2,y_2) \: \: is}$

$= \sf{ \sqrt{ {(x_2 - x_1)}^{2} + {(y_2 - y_1)}^{2} } \: }$

CALCULATION

Let the ordinate of the point P is k

Then Abscissa of the point P is 2k

So the point P is ( 2k , k )

Now the distance of the point P from the point (3, 4)

$= \sf{ \sqrt{ {(2k - 3)}^{2} + {(k - 4)}^{2} } \: }$

By the given condition

$\sf{ \sqrt{ {(2k - 3)}^{2} + {(k - 4)}^{2} } \: = \sqrt{10} }$

Squaring both sides we get

$\sf{ {(2k - 3)}^{2} + {(k - 4)}^{2} \: = 10}$

$\implies \: \sf{ 4 {k}^{2} - 12k + 9 + {k}^{2} - 8k + 16 = 10 }$

$\implies \: \sf{ 5 {k}^{2} - 20k + 15 = 0 }$

$\implies \: \sf{ {k}^{2} - 4k + 3 = 0 }$

$\implies \: \sf{ {k}^{2} - 3k - k + 3 = 0 }$

$\implies \: \sf{ k(k - 3) - 1(k - 3) = 0 }$

$\implies \: \sf{ (k - 3) (k - 1) = 0 }$

$\implies \: \sf{ k = 1 \: , \: 3 }$

For k = 1 the point is ( 2 , 1 )

For k = 3 the point is ( 6 , 3 )

RESULT

Hence the required points are (2 , 1 ), ( 6, 3)

━━━━━━━━━━━━━━━━

LEARN MORE FROM BRAINLY

Find the direction cosines of a line which makes equal angles with the coordinate axes

https://brainly.in/question/1508288