The variable chord on the parabola with equation subtends a right angle at the origin . By taking as and as , find a relation between and and hence show that passes through a fixed point on the .
If you've learnt vectors, you can say that :
Then, you'll have a relation between and
To find the fixed point, on the x-axis, find the equation of (PQ) (form y=ax+b) and solve it for the point of 0-ordinate (on the x-axis, all points have 0-ordinate)
Since these 2 vectors subtend a right angle the dot product of the vectors must be zero:
The first case happens if P = O or Q = O.
From the 2nd case we get: . Therefore the coordinates of the point Q become:
I only write t if is meant.
Calculate the equation of the line PQ:
Solve for y.
Now choose 2 different values for t, for instance t = r or t = s. You'll get 2 different equations for 2 different lines. Calculate the coordinates of the point of intersection between these 2 lines.
You'll get the Point I(4, 0)
I've attached a sketch of the situation.