SOLUTION: Common tangents are drawn to parabola y²=4x and ellipse 3x²+8y²=48 touching the parabola at A & B and the ellipse at C & D.Find the area of quadrilateral

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: Common tangents are drawn to parabola y²=4x and ellipse 3x²+8y²=48 touching the parabola at A & B and the ellipse at C & D.Find the area of quadrilateral      Log On


   

Question 1149909: Common tangents are drawn to parabola y²=4x and ellipse 3x²+8y²=48 touching the parabola at A & B and the ellipse at C & D.Find the area of quadrilateral
Answer by Edwin McCravy(20059) About Me  (Show Source):
You can put this solution on YOUR website!


We must find all the points of tangency. 

I'll assume you haven't had calculus, it's a lots harder that way! lol. 

Let the common tangent line have equation y=mx+b, solve the system

system%28y=mx%2Bb%2Cy%5E2=4x%29 
and find the discriminant of the resulting quadratic to be 1-bm. Set it equal to
0, because there is only one solution at a tangent point.  Then solve the
system:
system%28y=mx%2Bb%2C3x%5E2%2B8y%5E2=48%29 
and find the discriminant of the resulting quadratic to be -b²+16m²+6.  Set it
equal to 0.  Then solve the system
system%281-bm=0%2C-b%5E2%2B16m%5E2%2B6=0%29
You'll get


Substitute the positive values for b and m in y=mx+b, and solve simultaneously
with the parabola and get: 
A=%28matrix%281%2C3%2C8%2C%22%2C%22%2C4sqrt%282%29%29%29  
and you can tell by symmetry that 
B=%28matrix%281%2C3%2C8%2C%22%2C%22%2C-4sqrt%282%29%29%29
and with the ellipse and get:
A=%28matrix%281%2C3%2C-2%2C%22%2C%22%2C3sqrt%282%29%2F2%29%29  
and you can tell by symmetry that 
B=%28matrix%281%2C3%2C-2%2C%22%2C%22%2C-3sqrt%282%29%2F2%29%29

The quadrilateral CDBA is a trapezoid (or trapezium if you live in the UK).

Area=expr%281%2F2%29%28b%5B1%5D%2Bb%5B2%5D%29h

b%5B1%5D=6sqrt%282%29%2F2=3sqrt%282%29
b%5B2%5D=8sqrt%282%29
The height is along the x-axis from -2 to 8 which is 10 units.

Area=expr%281%2F2%29%283sqrt%282%29%2B8sqrt%282%29%29%2A10=55sqrt%282%29

It's much easier If you've had calculus. Just equate the derivatives to find the
values of x where the slopes are the same.  You'll get the same answer.

Edwin