Consider a fixed point (0,-2) and a fixed line y=2 . Verify...

Consider a fixed point (0,-2) and a fixed line $y=2 .$ Verify that the distance from each point to $(0,-2),$ is equal to the distance from the point to the line $y=2$ $P(12,-18) \quad Q\left(6,-\frac{9}{2}\right) \quad R(4 \sqrt{5},-10)$ $S(4 \sqrt{6},-12)$

Key Concepts

Distance Formula

The distance formula is used to calculate the distance between two points in the coordinate plane. Given two points, (x?, y?) and (x?, y?), the distance between them is found by the formula ?[(x? - x?)² + (y? - y?)²]. This formula is essential when determining how far a point is from a fixed point, as it systematically accounts for both horizontal and vertical displacements.

Distance from a Point to a Line

Finding the distance from a point to a line involves measuring the shortest distance between the point and the line, which is along the perpendicular dropped from the point to the line. In cases where the line is horizontal (or vertical), this distance simplifies to the absolute difference between the respective coordinate of the point and the constant value in the equation of the line. This concept is crucial when comparing distances in problems with a directrix.

Parabola (Focus-Directrix Definition)

A parabola is the set of all points in the plane that are equidistant from a fixed point (called the focus) and a fixed line (called the directrix). This equidistance property naturally leads to an equation that models the parabola. Verifying that a point is on the parabola involves checking that the distance from the point to the focus equals its distance to the directrix. This definition is the core concept underlying the problem statement.

Transcript

00:01 In this question we have to find distance from different points to points 0 comma minus 2 and line y equals to this equation of line we can rewrite as y minus 2 equals to 0.

00:21 Now first we will find distance between point p which is 12.

00:28 Minus 18 and point 0.

00:35 Distance is under root 0 minus 12 whole square plus bracket minus 2 minus bracket bracket minus 18 whole square on solving it we get the value of t which is 20 now we will find distance between point p 12.

01:03 Minus 18 and line by minus 2 equals to 0 we have t equals to mod 0 minus 18 minus 2 divided by under root 0 squared plus 1 square.

01:29 On solving it we get the value of t which is 20.

01:36 Hence it is verified that point p has an equal distance from point 0.

01:43 Minus 2 and line y equals to 2.

01:59 Now we will find distance between point q which is 6 comma minus 9 by 2 and point 0.

02:11 Minus 2.

02:14 So the distance is under root 0 minus 6 whole square plus minus 2 minus bracket minus 9 by 2 all square.

02:37 On solving it we get the value of the x x x all square.

02:41 As 13 by 2.

02:46 Now we will find distance between point q which is 6 comma 9 by 2 and line y minus 2 equals to 0.

03:04 So the formula is b equals to mod 0 plus minus 9 by 2 minus 2 divided by under root 0 square plus 1 square.

03:23 On solving it, we get the value 13 by 2.

03:26 Hence, it is verified that point q has an equal distance from 0, minus 2, and line y equals 2.

03:47 Now we will find distance between point r, which is 4 root 5, 0...

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