there are two lines through the given point, which are tangent to the given curve. Find the equation each of these lines.

4x^2- 5xy + 2y^2 + 3x - 2y = 0 , P ( 2,3)

need assistance on this please.

thank you

Printable View

- Jul 30th 2012, 11:26 PMrcsfind the equation of the lines.
there are two lines through the given point, which are tangent to the given curve. Find the equation each of these lines.

4x^2- 5xy + 2y^2 + 3x - 2y = 0 , P ( 2,3)

need assistance on this please.

thank you - Aug 1st 2012, 10:03 AMearbothRe: find the equation of the lines.
1. All lines passing through P have the equation: $\displaystyle y = m(x-2)+3$

2. Replace the term y in your equation of the ellipse by m(x-2)+3. You'll get a quadratic in x:

$\displaystyle x^2(2m^2 - 5m + 4) - x(8m^2 - 20·m + 12) + 4(2m^2 - 5·m + 3) = 0$

Solve for x.

3. Usually a straight line intercepts an ellipse in 2 or in 1 point or it is a passante that means there are no common points. The case that there exists only one point of interception occurs if the straight line is a tangent to the ellipse. This will happen if the discriminant equals zero:

$\displaystyle \sqrt{- 2·m^2 + 5·m - 3 - 2·m^2 + 5·m - 3)} = 0$

Solve for m. You should come out with $\displaystyle m = 1~\vee~m=\frac32$

4. Plug in these values into the equation of the line. - Aug 2nd 2012, 01:53 AMrcsRe: find the equation of the lines.
thank you for enlightening my brain... millions of thanks sir.

God Bless - Sep 3rd 2012, 07:04 PMrcsRe: find the equation of the lines.
- Sep 3rd 2012, 10:34 PMearbothRe: find the equation of the lines.
The marked equation describes

**an**ellipse, not**the**ellipse.

From the attachment of my first post you can see that you have a "tilted" ellipse, that means the axes are not parallel to the coordinate axes, but the curve is still an ellipse. - Sep 4th 2012, 04:16 AMrcsRe: find the equation of the lines.
i got it sir... i solved it manually and i was able to get that equation equal to 0. something that stuck me how you got square root (-2m^2 +5m -3 -2m^2 +5m - 3) = 0... is that the discriminant sir? which being plugged in by the values of the equation above?

thanks