# Math Help - Coordinate Systems

1. ## Coordinate Systems

The variable chord PQ on the parabola with equation y^2 = 4x subtends a right angle at the origin O. By taking P as (t₁^2, 2t₁) and Q as (t₂^2, 2t₂), find a relation between t₁ and t₂ and hence show that PQ passes through a fixed point on the x-axis.

2. Originally Posted by geton
The variable chord PQ on the parabola with equation y^2 = 4x subtends a right angle at the origin O. By taking P as (t₁^2, 2t₁) and Q as (t₂^2, 2t₂), find a relation between t₁ and t₂ and hence show that PQ passes through a fixed point on the x-axis.

Let O be the origin then we have,

$(\text{slope})_{OP}\cdot (\text{slope})_{OQ} = -1$

$\frac{2t_1 - 0}{t_1 ^2 - 0}\cdot \frac{2t_2 - 0}{t_2 ^2 - 0} = -1$

$\color{blue}t_1 t_2 = -4$

Now the equation of the line passing through $(t_1 ^2, 2t_1)$ and $(t_2 ^2, 2t_2)$ is given by $(t_1 + t_2)y = 2x + 2t_2t_2$.

Substitute $\color{blue}t_1 t_2 = -4$ to get $(t_1 + t_2)y = 2x - 8$.

Clearly this line always passes through (4,0) which is on the x-axis.

3. Originally Posted by Isomorphism
Let O be the origin then we have,

$(\text{slope})_{OP}\cdot (\text{slope})_{OQ} = -1$
Thank you for your help. But I’ve confusion. I know that product of tangent & normal is equal to -1. So why we suppose to assume OP is tangent & OQ is normal or vice-versa?

4. Originally Posted by geton
Thank you for your help. But I’ve confusion. I know that product of tangent & normal is equal to -1. So why we suppose to assume OP is tangent & OQ is normal or vice-versa?
No.I did not say they are tangent and normal.

If two lines OP and OQ are perpendicular, then

Since your question claimed "The variable chord PQ on the parabola with equation y^2 = 4x subtends a right angle at the origin O", OP and OQ are perpendicular.

5. Thank you so much Isomorphism