Find N And R If The Ratio Of Combinations ^nC_r : ^nC_{r+1} : ^nC_{r+2} Is 1:2:3. Determine The General Equation Of The Parabola With Vertex At The Origin Passing Through (-2, -4) And (-2, 4).

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Delving into the realm of combinatorics, we encounter the fascinating world of combinations, where the order of selection doesn't matter. The notation ^nC_r represents the number of ways to choose r items from a set of n distinct items. This concept is fundamental in various fields, from probability theory to computer science.

When faced with ratios of combinations, a unique opportunity arises to unravel the underlying values of n and r. Let's embark on a journey to decipher the given ratio: ^nC_r : ^nC_r+1} ^nC_{r+2 = 1:2:3. Our mission is to determine the specific values of n and r that satisfy this intriguing relationship.

To conquer this challenge, we'll leverage the fundamental formula for combinations: ^nC_r = n! / (r! * (n-r)!), where n! denotes the factorial of n, the product of all positive integers up to n. By applying this formula and manipulating the ratios, we'll unlock the secrets hidden within the combinatorial expressions.

First, let's express the given ratios as equations:

  1. ^nC_r / ^nC_{r+1} = 1/2
  2. ^nC_{r+1} / ^nC_{r+2} = 2/3

Now, let's substitute the combination formula into these equations. For the first equation, we have:

[n! / (r! * (n-r)!)] / [n! / ((r+1)! * (n-r-1)!)] = 1/2

Simplifying this expression, we get:

(r+1) / (n-r) = 1/2

Cross-multiplying, we arrive at:

2(r+1) = n-r

2r + 2 = n - r

n = 3r + 2

Next, let's tackle the second equation:

[n! / ((r+1)! * (n-r-1)!)] / [n! / ((r+2)! * (n-r-2)!)] = 2/3

Simplifying, we obtain:

(r+2) / (n-r-1) = 2/3

Cross-multiplying yields:

3(r+2) = 2(n-r-1)

3r + 6 = 2n - 2r - 2

2n = 5r + 8

Now, we have two equations with two unknowns:

  1. n = 3r + 2
  2. 2n = 5r + 8

We can substitute the first equation into the second equation to eliminate n:

2(3r + 2) = 5r + 8

6r + 4 = 5r + 8

r = 4

With the value of r in hand, we can substitute it back into the first equation to find n:

n = 3(4) + 2

n = 14

Thus, we have successfully deciphered the values of n and r: n = 14 and r = 4. This solution elegantly satisfies the given ratio of combinations, showcasing the power of combinatorial principles.

Transitioning from combinations to the realm of conic sections, we encounter the graceful parabola. A parabola is defined as the set of all points equidistant to a fixed point, the focus, and a fixed line, the directrix. Its symmetrical U-shaped curve graces numerous applications, from satellite dishes to projectile trajectories.

Our next challenge involves finding the general equation of a parabola with a specific set of characteristics: its vertex resides at the origin, and it gracefully passes through the points (-2, -4) and (-2, 4). These constraints provide valuable clues that will guide us towards the desired equation.

The general equation of a parabola with its vertex at the origin can take two primary forms:

  1. x^2 = 4ay, if the parabola opens upwards or downwards.
  2. y^2 = 4ax, if the parabola opens to the right or left.

To determine the appropriate form, we'll analyze the given points. Notice that both points, (-2, -4) and (-2, 4), share the same x-coordinate. This implies that the parabola opens either upwards or downwards, making the first form, x^2 = 4ay, the suitable choice.

Now, our mission is to find the value of a, which dictates the parabola's curvature. We can leverage the given points to achieve this. Let's substitute the coordinates of one of the points, say (-2, 4), into the equation x^2 = 4ay:

(-2)^2 = 4a(4)

4 = 16a

a = 1/4

With the value of a determined, we can now write the general equation of the parabola:

x^2 = 4(1/4)y

x^2 = y

Therefore, the general equation of the parabola that satisfies the given conditions is x^2 = y. This elegant equation encapsulates the parabola's shape and position, a testament to the beauty of mathematical relationships.

In this exploration, we've traversed the realms of combinatorics and conic sections, unraveling the mysteries of combinations and parabolas. We successfully determined the values of n and r that satisfy a given ratio of combinations, and we derived the general equation of a parabola that gracefully passes through specified points.

These mathematical endeavors not only showcase the power of formulas and equations but also highlight the elegance and interconnectedness of mathematical concepts. By mastering these principles, we equip ourselves to tackle a wide array of challenges in mathematics and beyond, fostering a deeper appreciation for the beauty and utility of this fundamental discipline.