Respuesta :
Answer:
- a. (0,0)
- c. (−4,3)
- d. (16,8)
Step-by-step explanation:
You want the points at the foot of normals to the given parabola from point (7, 14).
Approach
The approach here is to write expressions for the slope of a normal to the curve, and for the slope from (7, 14) to a point on the curve. We then evaluate the difference of these slopes for the offered points. Since we know the offered points are all on the parabola, any that make the slope difference zero will be a point at the foot of a normal.
Normal
The slope of a normal will be the opposite reciprocal of the slope of the parabola at x. The parabola's slope is the derivative of the function at that point:
y = (x² -8x)/16
y' = (2x -8)/16 = (x -4)/8
The slope of the normal at point (x, y) on the parabola is the opposite reciprocal of this:
m = 8/(4 -x)
The slope from point (7, 14) to point (x, y) on the curve is ...
m = (y -14)/(x -7)
For a point (x, y) at the foot of a normal, both these slopes must be the same. We have rewritten the equation so each variable appears only once, making calculator evaluation somewhat simpler.
[tex]\dfrac{(y -14)}{(x -7)} = \dfrac{8}{(4 -x)}\qquad\text{line slope is normal slope}\\\\\\\dfrac{y-14}{8}-\dfrac{x-7}{4-x}=0\qquad\text{multiply by (x-7)/8 and subtract right side}\\\\\\\dfrac{y-14}{8}-\dfrac{3}{x-4}+1=0[/tex]
A calculator can help us figure which of the given choices satisfies this equation.
Feet
The attached calculator display shows the equation for the slopes is satisfied for points ...
A(0, 0), C(-4, 3), D(16, 8)
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Additional comment
The vertex of the parabola is (4, -1), where the normal is a vertical line. The point (7, 14) has a different x-coordinate, so is not on that line. The slope of the vertical line gives the result "complex infinity" when the calculator tries to compute it.
If we write an equation for the x-values of the feet of the normals, we find it is a cubic. It has solutions x ∈ {-4, 0, 16}, for which the equation of the parabola tells us corresponding y-values are {3, 0, 8}.
Final answer:
In High School Mathematics, the foot of the normal from the point (7, 14) to the parabola represented by x² − 8x − 16y = 0 is found by rewriting the equation in standard form and applying the concept of normals to parabolas. After calculation, the correct coordinate for the foot of the normal is (4, -1). So, the correct option is B.
Explanation:
The subject of this question is Mathematics, specifically relating to the geometrical properties of a parabola and the concept of normals to parabolas. To find the foot of the normal from the point (7, 14) to the parabola given by the equation x² − 8x − 16y = 0, we need to rewrite the equation in its standard form and then use the properties of normals to such a curve.
The given parabola equation can be rearranged by completing the square to find its vertex form. Completing the square, we get (x-4)² = 16y. Thus the vertex of the parabola is (4, 0) and the focus is (4, 1). Normals to a parabola from an external point are straight lines, and for a parabola of the form y² = 4qx, normals at point (at², 2at) have a particular slope, and that formula can be used to find the normal that passes through a given external point. Hence, we calculate the possible coordinates for the foot of the normal.
After comparing the slopes and calculating the equations of the potential normals, we can determine that the correct option from the given choices is (4, −1), which is option b. This calculation involves the use of algebraic manipulation and understanding of the geometric properties of the parabola. It's important to mention the correct option in the final part so the student can confirm their answer.
