coordinate geometry

• Aug 20th 2010, 06:40 AM
hooke
coordinate geometry
Show that the tangent to the ellipse ax^2+by^2=1 at the point (h,k) has equation ahx+bky=1

Hence, deduce that the chord of contact of tangents from the point (m,n) to the ellipse ax^2+by^2=1 has equation amx+bny=1

I can do the 2 questions above.

Show that for all values of t, the chord of contact of tangents from the point (2t, 1-t) to the ellipse ax^2+by^2=1 passes through a fixed point and determine the point of this coordinates.

I am only stuck with this.
• Aug 20th 2010, 07:20 AM
Opalg
Quote:

Originally Posted by hooke
Show that the tangent to the ellipse ax^2+by^2=1 at the point (h,k) has equation ahx+bky=1

Hence, deduce that the chord of contact of tangents from the point (m,n) to the ellipse ax^2+by^2=1 has equation amx+bny=1

I can do the 2 questions above.

Show that for all values of t, the chord of contact of tangents from the point (2t, 1-t) to the ellipse ax^2+by^2=1 passes through a fixed point and determine the point of this coordinates.

I am only stuck with this.

Put m = 2t and n = 1 – t in the equation of the chord: \$\displaystyle 2tax + (1-t)by = 1\$. Write that as \$\displaystyle t(2ax-by) = 1-by\$. If that holds for all t then \$\displaystyle 2ax-by = 1-by = 0\$. Solve those equations to find the values of x and y for the fixed point.