locus of an ellipse q (1 Viewer)

[bally24]

point p divides a line AB in the ratio of 1:2 where A is a point on the yaxis and B is a point on the x axis.
i) show that the locus of P is an ellipse and write its equation.
ii) what would the locus of P be if P were midpoint of AB?

q from textbook...no answer, thanks for ur help!

 

[Riviet]

I believe that the locus problems with ellipses are not in the syllabus, therefore they are not examinable in the HSC. However, they may be in your trials, as your teacher might put one in it.

My thoughts on the question: Let the points be represented by A(0,y₁) and B(x₁,0). Use the ratio division formula to derive two expressions. Then try to find a relationship between the two expressions and try to come up with something that resembles the equation of an ellipse.

For part ii), if P is the midpoint, then it would divide AB in the ratio 1:1, so try using ratio division formula again and apply a similar method like in the first part. I'm guessing the locus is a circle.

 

[shimmerz_777]

wouldnt part two be a parabola if its the midpoint, as the ecentricity is 1?

 

[Riviet]

Ah yes, you're right, totally forgot about the parabola.