Parametrics question (1 Viewer)

[HeroWise]

P is any variable point on the parabola x^2=-4y. THe tangent from P cuts the parabola x^2=4y at Q and R. Show that 3x^2=4y is the equation of the licus of the mid point of the chord RQ.






PS: How do you activate LateX editor??

 

[fan96]

use [ tex ] and [ /tex ] (remove the spaces inside the square brackets).

For example, [ tex ] x [ /tex ] gives

The equation of the tangent at any point on the parabola is given by:



Solving simultaneously the equations of the tangent and the parabola , we get



Treating this as a quadratic in , we can solve it using the quadratic formula to obtain:





Taking the midpoint and simplifying gives us the parametric equation



and it's easy to show that the equivalent Cartesian equation is .

(A nice trick you can use to find the midpoint in questions like these is to halve the sum of roots - this is most useful when you don't need the coordinates.)