Hey everyone! Today, we're diving deep into the fascinating world of parabolas and tackling some intriguing problems. We'll be focusing on two key concepts: normals to a parabola and poles of straight lines with respect to a parabola. So, buckle up and let's get started!
1. Finding the Value of k When a Line is Normal to a Parabola
Let's kick things off with our first problem: If the straight line x + y = k is normal to the parabola y² = 12x, then what's the value of k? This is a classic problem that combines the concepts of straight lines and parabolas, and it's a great way to test your understanding of these topics.
Understanding Normals to a Parabola
First, let's quickly recap what a normal to a parabola is. A normal is simply a line that is perpendicular to the tangent at a given point on the parabola. Think of it as the line that points directly away from the curve at that specific point.
To solve this problem, we need to find the condition for a line to be normal to the given parabola. There are a couple of ways we can approach this, but let's go with the method that involves the equation of the normal in terms of the slope.
The Equation of the Normal
For the parabola y² = 4ax, the equation of the normal with slope 'm' is given by:
y = mx - 2am - am³
This formula is a powerful tool for solving problems related to normals, so it's worth memorizing! In our case, the given parabola is y² = 12x, which means 4a = 12, and therefore a = 3.
Transforming the Given Line
Now, let's rewrite the equation of the given line, x + y = k, in the slope-intercept form (y = mx + c). This gives us:
y = -x + k
From this, we can see that the slope of the line, m, is -1. This is a crucial piece of information that we'll use in the next step.
Applying the Normal Condition
Now comes the fun part! We'll substitute the values of 'a' and 'm' into the equation of the normal:
y = (-1)x - 2(3)(-1) - (3)(-1)³ y = -x + 6 + 3 y = -x + 9
Notice that this equation represents the normal to the parabola with a slope of -1. But we also know that the given line, y = -x + k, is normal to the parabola. For these two lines to be the same, their y-intercepts must be equal.
Finding the Value of k
Comparing the equations y = -x + 9 and y = -x + k, we can clearly see that k = 9.
Therefore, the correct answer is (d) 9.
Isn't it cool how we used the equation of the normal and a little bit of algebraic manipulation to solve this problem? These kinds of problems really highlight the beauty and interconnectedness of different mathematical concepts.
2. Decoding Poles and Polars: Finding the Pole of a Line
Alright, let's move on to our second problem: Coordinates of the pole of the straight line 2x - 5y - 4 = 0 with respect to the parabola y² = 8x are what? This problem introduces us to the concepts of poles and polars, which are fascinating geometric relationships between points and lines with respect to conic sections like parabolas.
Understanding Poles and Polars
Before we dive into the solution, let's get a clear understanding of what poles and polars are. Imagine you have a parabola and a point (let's call it P) outside the parabola. If you draw two tangents from point P to the parabola, the line joining the points of contact of these tangents is called the polar of point P with respect to the parabola. Conversely, the point P is called the pole of that line.
Think of it as a special relationship: every point outside the parabola has a corresponding line (its polar), and every line has a corresponding point (its pole).
The Equation of the Polar
For the parabola y² = 4ax, the equation of the polar of a point (x₁, y₁) is given by:
yy₁ = 2a(x + x₁)
This is another important formula to remember when dealing with poles and polars. It allows us to find the equation of the polar if we know the point, or vice versa.
Applying the Concept to Our Problem
In our problem, we're given the equation of the polar (2x - 5y - 4 = 0) and the equation of the parabola (y² = 8x). Our goal is to find the coordinates of the pole (x₁, y₁).
First, let's rewrite the equation of the parabola in the standard form, y² = 4ax. Comparing y² = 8x with y² = 4ax, we get 4a = 8, which means a = 2.
Now, let's assume the coordinates of the pole are (x₁, y₁). Using the formula for the polar, we can write the equation of the polar as:
yy₁ = 2(2)(x + x₁) yy₁ = 4(x + x₁) yy₁ = 4x + 4x₁
Comparing Equations and Solving for the Pole
We now have two equations representing the same line:
- yy₁ = 4x + 4x₁
- 2x - 5y - 4 = 0
To compare these equations, let's rewrite the second equation in a form that looks more like the first equation. We can multiply the second equation by a constant, say λ (lambda), to make the coefficients comparable:
λ(2x - 5y - 4) = 0 2λx - 5λy - 4λ = 0
Now, let's rearrange this equation to match the form of the first equation:
-5λy = -2λx + 4λ
Comparing the coefficients of x, y, and the constant terms in the two equations, we get the following relationships:
- Coefficient of x: 4 = -2λ => λ = -2
- Coefficient of y: y₁ = -5λ => y₁ = -5(-2) = 10
- Constant term: 4x₁ = -4λ => 4x₁ = -4(-2) => x₁ = 2
Oops! It seems we made a small error in our rearrangement. Let's go back and correct it. The correct rearrangement of the second equation should be:
5λy = 2λx - 4λ
Now, comparing the coefficients, we get:
- Coefficient of x: 4 = 2λ => λ = 2
- Coefficient of y: y₁ = 5λ => y₁ = 5(2) = 10
- Constant term: 4x₁ = -4λ => 4x₁ = -4(2) => x₁ = -2
The Grand Finale: The Coordinates of the Pole
So, after carefully comparing the equations and solving for x₁ and y₁, we find that the coordinates of the pole are (-2, 10).
Therefore, the correct answer is (a) (-2, 10).
This problem beautifully illustrates the power of using the polar equation and comparing coefficients to solve geometric problems. It's like cracking a code, isn't it?
Conclusion: Mastering Parabolas and Their Properties
Guys, we've covered a lot of ground today! We've tackled problems involving normals to parabolas and poles of lines with respect to parabolas. These are fundamental concepts in conic sections, and understanding them will definitely boost your problem-solving skills.
Remember, the key to mastering these concepts is to:
- Understand the definitions: Make sure you have a solid grasp of what normals, poles, and polars are.
- Memorize the formulas: The equations for the normal and the polar are your best friends in these types of problems.
- Practice, practice, practice: The more problems you solve, the more comfortable you'll become with applying these concepts.
So, keep practicing, keep exploring, and keep the math magic flowing! Until next time, happy problem-solving!
