thinay Posts: 40, Reputation: 2 Junior Member #1 Sep 13, 2009, 07:08 AM
Problems in tangents and normals
Please help me find the solutions in the problems below. I will really appreciate any help. Here it goes...

In the first problem, find the equation of the line to the ff. conics at the given point.

1.) 3x^2 - 3xy + 4x + y - 3 = 0
given point: (-1,1)
Answer: 5x - 4y + 9 = 0

Next problem, find the equations of the lines which are tangent to the to the ff. conics and pass through the given point.

1.) x^2 + 2xy + y^2 + 2x + 6y + 1 = 0
given point: (4,-2)
Answer: x + 2y = 0, 2x + 3y = 2

Hope someone can help me in here. Thanks in advance! :)
 galactus Posts: 2,271, Reputation: 282 Ultra Member #2 Sep 13, 2009, 08:32 AM
In the first problem, find the equation of the line to the ff. conics at the given point.

1.) 3x^2 - 3xy + 4x + y - 3 = 0
given point: (-1,1)
Answer: 5x - 4y + 9 = 0
Solve the equation for y and get $y=\frac{3x^{2}+4x-3}{3x-1}$

Find its derivative in order to find the slope at the point x=-1.

$\frac{dy}{dx}=\frac{9x^{2}-6x+5}{(3x-1)^{2}}$

Plug in x=-1 and we find the slope at that point is m=5/4.

Now, use the slope intercept form of the line equation, y=mx+b

You now know x, y, and m. Plug them in and solve for b.

Then, merely algebraically transform it in the standard form of the line equation if need be.

If done correctly, you will get the answer you have posted.
 thinay Posts: 40, Reputation: 2 Junior Member #3 Sep 17, 2009, 08:43 AM

Ok I got it! Thanks! But how about in the next problem? Wherein there are 2 answers? :(

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