conic questions :( (1 Viewer)

[coeyz]

1.) Write both Carterian equation and the parametric equation for the conic
which has foci (-6, -2) and (4, -2), and an eccentricity of 5/4

ans: [(x+1)^2]/25 - [16(y+2)^2]/225 = 1
x = 5sec日-1 y=(15/4)tan日=2

2.) Find the equation of the normal to the ellipse x^2 /100 + y^2 / 75 = 1 at the point P (5, 7.5) < ye i can do this
but next question :
Find the equation of the circle that is tangential to the ellipse at P and Q(5, 7.5)
What does it mean@@@@@@@?

THANKSSSSSSSSSSS

 

[IAU001]

hey quoted solution by cyl123 from:
http://community.boredofstudies.org.../conics-half-yearly-past-paper-questions.html

tangential means the circle TOUCHES the ellipse at that point and share the same tangent

the easiest way i can think of to do the question is to find the normal of the ellipse at both points. Since the circle and ellipse share the same tangents, they share the same normal. knowing that normals to circle pass thru centre,, this can be used to find centre and then radius of circle

So the normal to ellipse (and circle) at (5,7.5) is 4x-2y=5
the normal at (5, -7.5) is 4x+2y=5
Since radius is perpendicular to tangent of circle, which is shared by ellipse, then the radius must be the normal to the ellipse, so where the 2 normals meet will be the centre.

Solving simulataneously gives x = 5/4, y=0

And radius is found using distance formula to be sqrt(70.3125)

So equation of circle is (x-5/4)^2+y^2 = 70.3125

sorry thats only for question 2. ill try and do question 1 for u if i can!!