Parabola!!

For the parabola $x^2+4y^2-4xy-10x+5=0$ , find the equation of tangent at vertex and equation of axis of parabola?

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

This is an interesting problem, namely because the parabola's axis is not parallel to an x-y axis. We are used to dealing with conics in this form. However, using the definition of a parabola (distance to a point on the parabola is the same from a fixed point called the focus and a fixed line called the directrix), one can develop a hairy formula for a general parabola. Plugging in and solving for some variables, I got a possible focus to be (1,0) and a possible directrix of x-2y=0. The axis of the parabola then must be 2x+y=2, and the vertex (9/10,7/10). There could be more answers, but I didn't check. If you want me to look more into this problem or write a formal proof, I'll consider it.

Bob Krueger - 8 years ago

your in level four of geometry...please give it a try at least....it does not need much perspiration...best of luck ....:)

Sayan Chaudhuri - 8 years ago

Dude this is already done... but i liked this problem very much that's why posted that..

Advitiya Brijesh - 8 years ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$