Conics Q... (1 Viewer)

[h4vok]

Prove that the condition for the line y = mx + c to be a tangent to the ellipse x^2/a^2 + y^2/b^2 = 1 is c^2 = a^2 m^2 + b^2.

Hence prove that the pair of tangents from P(4,5) to the ellipse x^2/25 + y2/16 = 1 are at right angles to one another.

I have no idea..

 

[Mill]

The answer to this is long and I don't want to type it. It also would not be an enriching and fulfiliing experience for you if I did so.

That said, I will point you in the right direction (ie. tell you the answer without writing it out ).

Solve the two equations simultaneously (Hint: substitute y = mx + c into the equation for the ellipse). Rearrange to gain a quadratic in x.

Now, you want the line to be tangential to the curve (ie. to TOUCH it but NOT to cross it). So you want only one solution to this quadratic equation you have formed.

So let the discrimnant of your quadratic equation equal zero and you will obtain the solution you are looking for!

 

[Roobs]

Ive been doing the exact same question, and i saw an alternative method in S.K Patel's Extention 2 exel book.

Page 156 for those who have the book-- it involves "comparing the coefficents"of y=mx+k with those of the general from of the tangent to an ellipse:

eg: mx-y=k and xx1/a^2+yy1/b^2=1

"then comparing coefficients" : (x1/a^2)/m = (y1/b^2)/-1 = 1/k

and then solving for x1 and y1, and substituting them into the ellipse ( as( x1, y1) is the tangents point of contact) to obtain something close to the condition required-- namely k^2=a^2m^2+ b^2

Is this a valid method, it seems to work -- though i havent been taught or heard about it, and i cant seem to understand the logic behind it-- anyone who can shed any light onto this, it is greatly appreciated.....i'd ask my teacher but i got a test tommorow...lol