The focus of the parabola is (-1, -9/4), and the directrix is the line y = -7/4. Show that the point (-2,-3) on the parabola is the same distance from the focus as from the directrix.
Well, I do not know what the parabola has to do with this.
We are asked the distance from
(-2,3) to (-1, -9/4)
and from that same point
(-2,3) to the horizontal line y = -7/4
The second part is trivial
-7/4 to 3 is 12/4 + 7/4 = 19/4
we will actually have to do the first distance and hope it comes out 19/4
(-2,3) to (-1, -9/4)
d^2 = (-9/4 - 12/4)^2 + 1^2
d^2 = (-21/4)^2 + (4/4)^2
d^2 = (1/4^2) (441+16) = (1/4^2)(457)
d = (1/4) sqrt 457
hmmm, something is wrong
sqrt 457 is not 19
To show that the point (-2, -3) on the parabola is equidistant from the focus and the directrix, we can use the distance formula.
1. Calculate the distance between the point (-2, -3) and the focus (-1, -9/4).
- The distance between two points (x₁, y₁) and (x₂, y₂) is given by the formula: distance = √((x₂ - x₁)² + (y₂ - y₁)²).
- Substituting the values, the distance between (-2, -3) and (-1, -9/4) is: distance₁ = √((-1 - (-2))² + (-9/4 - (-3))²).
2. Calculate the distance between the point (-2, -3) and the directrix.
- Since the directrix is a horizontal line y = -7/4, the distance between the point (-2, -3) and the directrix is the vertical distance between the point and the line.
- The vertical distance between a point (x, y) and a horizontal line y = a is given by the formula: distance = |y - a|.
- Substituting the values, the distance between (-2, -3) and y = -7/4 is: distance₂ = |-3 - (-7/4)|.
3. Compare the distances.
- Compare the calculated distances: distance₁ and distance₂.
- If distance₁ = distance₂, then the point (-2, -3) is equidistant from the focus and the directrix.
4. Solve the equation.
- Calculate and compare distance₁ and distance₂.
- If distance₁ = distance₂, then (-2, -3) is equidistant from the focus and the directrix.
By following these steps and performing the calculations, you can determine whether the point (-2, -3) on the parabola is equidistant from the focus and the directrix.