Doubts in straight lines chapter

[Mathematist]

While solving questions in the chapter 'Straight lines' I came across some questions which I'm unable to solve.

These are the questions:

1. Find the equation of straight lines passing through (-2,-7) and having an intercept of length 3 units between the straight lines 4x + 3y = 12, 4x + 3y = 3.

2. A variable straight line passes through the points of intersection of the lines x + 2y = 1 and 2x - y = 1 and meets the co-ordinates axes in A and B. Prove that the locus of the midpoint of AB is 10xy = x + 3y.

3. A right angled triangle ABC having a right angle at C, CA = b and CB = a, move such that the angular points A and B slide along the x-axis and y-axis respectively. Find the locus of C.

4. The base of a triangle is axis of x and its other 2 sides are given by the equations :
y = (1+α)x/α + (1+α) and y = (1+β)x/β + (1+β).
Prove that the locus of its orthocentre is the line x+y = 0.

5. If the lines joining origin and the point of intersection of curves ax² + 2hxy + by² + 2gx = 0 and a₁x² + 2h₁xy + b₁y² + 2g₁x = 0 are mutually perpendicular, then prove that g(a₁ + b₁) = g₁(a+b).

What is my difficulty in these questions:

1. Tried whatever I could.. So please tell me how to solve it.

2. I can find the point of intersection but I don't know what to do after that.

3. What does angular points mean ?

4. Tried whatever I could.. So please tell me how to solve it.

5. I can see that the given curves meet at 2 points and (0,0) is one of them. Then, the questions says the lines joining the origin and the point of intersection of the given curves are perpendicular. So the point of intersection must be the point other than (0,0) which satisfies both the curves. Then the questions says that 2 lines are drawn such that they are perpendicular to each other and also one passes through the origin and the other through the other point of intersection. First I'd like to know if my understanding of the question is correct. If yes, then please tell me how to proceed to solve it because I couldn't get the answer after trying to solve it for more than 2hrs. If my understanding of the question is wrong then please tell me what the question actually says.

 

[stapel]

1. Find the equation of straight lines passing through (-2,-7) and having an intercept of length 3 units between the straight lines 4x + 3y = 12, 4x + 3y = 3.

My work: Tried whatever I could.. So please tell me how to solve it.

What did you try? What were your results? For instance, you started by noticing that these where two parallel lines and, in solving for "y=", you discovered that these parallel lines were... how many units (in height) apart from each other?

Please reply showing this, so we can help you find any errors (or -- who knows? -- maybe let you know that you're almost to the answer!).

2. A variable straight line passes through the points of intersection of the lines x + 2y = 1 and 2x - y = 1 and meets the co-ordinates axes in A and B. Prove that the locus of the midpoint of AB is 10xy = x + 3y.

My work: I can find the point of intersection but I don't know what to do after that.

Is "variable straight line" defined as "a straight line which moves across or through the stated environment"? So this line (let's call it "L") represents a class of lines, all of which (i) pass through the intersection point and (ii) pass through both of the coordinate axes?

3. A right angled triangle ABC having a right angle at C, CA = b and CB = a, move such that the angular points A and B slide along the x-axis and y-axis respectively. Find the locus of C.

My work: What does angular points mean ?

The grammar of the exercise is bad, so the meaning is unclear. (The subject is the singular "triangle", but the verb "move" is plural.) But, as near as I can tell, "angular points" are just vertices. So it's saying that you have a triangle where the right angle is at point C and the other angle vertices are on the x- and y-axes. I'm guessing that, implicit in the exercise statement, you're supposed to assume that C is not at the origin. (So: picture the right triangle with C off the axes.)

4. The base of a triangle is axis of x and its other 2 sides are given by the equations :
y = (1+α)x/α + (1+α) and y = (1+β)x/β + (1+β).
Prove that the locus of its orthocentre is the line x+y = 0.

My work: Tried whatever I could.. So please tell me how to solve it.

Please see my answer for (1) above.

5. If the lines joining origin and the point of intersection of curves ax² + 2hxy + by² + 2gx = 0 and a₁x² + 2h₁xy + b₁y² + 2g₁x = 0 are mutually perpendicular, then prove that g(a₁ + b₁) = g₁(a+b).

My work: I can see that the given curves meet at 2 points and (0,0) is one of them.

How are you "seeing" this?

Then, the questions says the lines joining the origin and the point of intersection of the given curves are perpendicular. So the point of intersection must be the point other than (0,0) which satisfies both the curves. Then the questions says that 2 lines are drawn such that they are perpendicular to each other and also one passes through the origin and the other through the other point of intersection. First I'd like to know if my understanding of the question is correct. If yes, then please tell me how to proceed to solve it because I couldn't get the answer after trying to solve it for more than 2hrs. If my understanding of the question is wrong then please tell me what the question actually says.

From the equations, we know that these are ellipses of some sort, because the by-themselves x- and y-terms are each squared and have "plus" signs on them. (If a = b, then they're circles.) In completing the squares (if we'd have the values for the various constants), presumably we'd find their centers, axis lengths, etc.

In imagining two overlapping ellipses, such that their intersection points are exactly two (rather than one: one vertex; or four: crossing "through" each other; or none: being completely distinct; etc), what sort of intersection do we "see"? Does this lead you anywhere useful?

 

[Mathematist]

1. Yes, I noticed that they were parallel and I also found the distance between them as 9/5. But now I don't know what to do... I tried assuming the eq. of the line as ax+by+c = 0 and substituted it on (-2,-7) to get the eq. as x(a+2) + y(b+7) = 0. I tried to find the coordinates of the points of intersection of this line with the parallel lines but seems it is getting complicated that way. Then I was thinking of other ways to solve it. But got none....

2. Yes I tried doing that. But after you find that a family/class of lines pass through that point, then what ? I can't think of any way to solve it....

3. Ok, I got the answer for this. It is bx +/- ay = 0.

4. I tried to think what ever I could but I couldn't find any way to arrive at the answer. I don't know what to do... Giving me the solution is just enough. I can learn from that. Or you can at least tell me the way to solve it.

5. Straight lines or curves concept can be used to solve this question. I don't know anything about ellipse and knowledge about that isn't needed for solving this question. And first of all, is my understanding of the question correct that 2 lines should be drawn such that they are perpendicular to each other and also one passes through the origin and the other through the other point of intersection ? If this is correct, then please the way to solve it. As I said earlier, I don't know how to do it and giving the solution or the way to solve it would be much helpful so that I can then move on to solving even more difficult questions or other chapters.

 

[Deleted member 4993]

1. Yes, I noticed that they were parallel and I also found the distance between them as 9/5. But now I don't know what to do... I tried assuming the eq. of the line as ax+by+c = 0 and substituted it on (-2,-7) to get the eq. as x(a+2) + y(b+7) = 0. I tried to find the coordinates of the points of intersection of this line with the parallel lines but seems it is getting complicated that way. Then I was thinking of other ways to solve it. But got none....

2. Yes I tried doing that. But after you find that a family/class of lines pass through that point, then what ? I can't think of any way to solve it....

3. Ok, I got the answer for this. It is bx +/- ay = 0.

4. I tried to think what ever I could but I couldn't find any way to arrive at the answer. I don't know what to do... Giving me the solution is just enough. I can learn from that. Or you can at least tell me the way to solve it.

5. Straight lines or curves concept can be used to solve this question. I don't know anything about ellipse and knowledge about that isn't needed for solving this question. And first of all, is my understanding of the question correct that 2 lines should be drawn such that they are perpendicular to each other and also one passes through the origin and the other through the other point of intersection ? If this is correct, then please the way to solve it. As I said earlier, I don't know how to do it and giving the solution or the way to solve it would be much helpful so that I can then move on to solving even more difficult questions or other chapters.

If draw a circle, centered at (-2,-7) with a radius of 3:

where would those two parallel lines intersect this circle?

 

[Mathematist]

If draw a circle, centered at (-2,-7) with a radius of 3:

where would those two parallel lines intersect this circle?

How is this supposed to lead to the answer ?

 

[Mathematist]

Btw, I got the answer to the 5th question by searching google .
It seems, once I transform one of the curves to homogeneous equation of a pair of straight lines, I easily get the answer.