Coordinate Geometry Q: Intersecting Lines & Parallelogram

[Harmony]

Question Statement
The straight line y=mx -2 intersects the curve y^2=4x at the two points P(x1,y1) and Q(x2,y2). Show that
1) m>-1/2, m not equal to 0.
2) x1 + x2 = 4(m+1)/m^2
3) y1 + y2 = 4/m

If the point O is the origin and the point T is a point such that OPTQ is a parallelogram, show that, when m changes, the equation of the locus of T is y^2 + 4y = 4x

My attempt:

The first part of the question is pretty straightforward, but the second part of the question (regarding the parallelogram) is a bit confusing.

My solution so far:
Let T be (x,y)
Gradient of OP = gradient of TQ

y1/x1 = y-y2 / x-x2
xy1 - x2y1 = yx1 - xy2
yx1 = xy1 + xy2 - x2y1...(1)

Similarly, Gradient of PT = gradient of OQ

yx2 = xy2 -xy2 + x2y1...(2)

(1) + (2)
(x1 + x2)y = (y1 + y2)x
4y(m+1)/ m^2 = 4x/m
y(m+1)/m = x

But I can't get rid of the m afterward

 

[matness]

If your parellogram is in the form OPTQ this means simply T is the sum of the vectors P and Q
From this we conclude T=(x1+x2,y1+y2)
and from the first part you know the values
just try the equation given to you whether T is on that or not

 

[Architeuthis Dux]

As a scientist, it is important to carefully read and analyze the given question statement before attempting to solve it. In this case, the question is asking to show that the equation of the locus of T, when m changes, is y^2 + 4y = 4x. This means that the equation is not dependent on m, and thus, we need to eliminate m in our solution.

To do this, we can use the equations we have derived for the points P and Q, which are y = mx - 2 and y^2 = 4x, respectively. We can substitute the value of x from the first equation into the second equation, giving us y^2 = 4(mx - 2). This can be simplified to y^2 = 4mx - 8.

Next, we can use the equation we derived for the locus of T, which is y(m+1)/m = x. We can rearrange this to get y = mx + x, and then substitute it into our simplified equation for the locus of T. This gives us (mx + x)^2 = 4mx - 8.

Expanding and simplifying this equation, we get y^2 + 4y = 4x, which is the desired equation for the locus of T. This shows that the locus of T is independent of m, and is a parabola with a vertex at (-2, -4).

In conclusion, as a scientist, it is important to carefully analyze and understand the given question statement and use logical reasoning and mathematical equations to solve it.