Question

The point on y-axis which is equidistant from the points (5, 4), (-2, 3) is
​

Answer 1

Step-by-step explanation:

midpoint = (x1+x2/2),(y1+y2/2)

= (5-2/2),(4+3/2)

= (3/2),(7/2)

=1.5,3.5

x axis =1.5

y axis=3.5

Answer 2

Answer:

The point on Y-axis which is equidistant from the given points is P ( 0, 14 ).

Step-by-step-explanation:

We have given the coordinates of two points.

We have to find the coordinates of a point which is on Y-axis and equidistant from the given points.

Let P be the point on Y-axis and equidistant from the given points.

$\bullet\:\sf\:A\:\equiv\:(\:5\:,\:4\:)\:\equiv\:(\:x_{1}\:,\:y_{1}\:)\\\\\bullet\:\sf\:B\:\equiv\:(\:-\:2\:,\:3\:)\:\equiv\:(\:x_{2}\:,\:y_{2}\:)\\\\\bullet\:\sf\:P\:\equiv\:(\:0\:,\:y\:)\:\equiv\:(\:x\:,\:y\:)$

As P is on Y-axis, its X-coordinate must be zero.

Now, A and B are equidistant from P.

$\therefore\sf\:d\:(\:A,\:P\:)\:=\:d\:(\:B,\:P\:)\\\\\implies\pink{\sf\:\sqrt{\:(\:x_{1}\:-\:x\:)^{2}\:+\:(\:y_{1}\:-\:y\:)^{2}}\:=\:\sqrt{\:(\:x_{2}\:-\:x\:)^{2}\:+\:(\:y_{2}\:-\:y\:)^{2}}}\sf\:\:\:-\:-\:[\:Distance\:\:formula\:]\\\\\implies\sf\:\sqrt{\:(\:5\:-\:0\:)^{2}\:+\:(\:4\:-\:y\:)^{2}\:}\:=\:\sqrt{\:[\:(\:-\:2\:)\:-\:0\:]^{2}\:+\:(\:3\:-\:y\:)^{2}\:}\\\\\implies\sf\:(\:5\:-\:0\:)^{2}\:+\:(\:4\:-\:y\:)^{2}\:=\:(\:-\:2\:)^{2}\:+\:(\:3\:-\:y\:)^{2}\:\:\:-\:-\:[\:Squaring\:both\:sides\:]\\\\\implies\sf\:(\:5\:)^{2}\:+\:(\:4\:)^{2}\:-\:2\:\times\:4\:\times\:y\:+\:(\:y\:)^{2}\:=\:(\:-\:2\:)^{2}\:+\:(\:3\:)^{2}\:-\:2\:\times\:3\:\times\:y\:+\:(\:-\:y\:)^{2}\\\\\implies\sf\:25\:+\:16\:-\:8y\:+\:\cancel{y^{2}}\:=\:4\:+\:9\:-\:6y\:+\:\cancel{y^{2}}\\\\\implies\sf\:41\:-\:8y\:=\:13\:-\:6y\\\\\implies\sf\:41\:-\:13\:=\:-\:6y\:+\:8y\\\\\implies\sf\:28\:=\:2y\\\\\implies\sf\:y\:=\:\cancel{\frac{28}{2}}\\\\\implies\boxed{\red{\sf\:y\:=\:14}}$

Additional Information:

1. Distance Formula:

The formula which is used to find the distance between two points using their coordinates is called distance formula.

$\large{\boxed{\red{\sf\:d\:(\:A\:,\:B\:)\:=\:\sqrt{\:(\:x_{1}\:-\:x_{2}\:)^{2}\:+\:(\:y_{1}\:-\:y_{2}\:)^{2}\:}}}\:\:}$

2. Section Formula:

The formula which is used to find the coordinates of a point which divides a line segment in a particular ratio is called section formula.

$\large{\boxed{\red{\sf\:x\:=\:\dfrac{mx_{2}\:+\:nx_{1}\:}{m\:+\:n}}}}\:\:\sf\:\&\:\:\:\large{\boxed{\red{\sf\:y\:=\:\dfrac{my_{2}\:+\:ny_{1}\:}{m\:+\:n}}}\:\:}$