Drsoccerball
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Can someone tell me what it is and how you use it? Its in ellipses.
If P is a point on an ellipse, then the normal to the ellipse at P bisects SPS', where S and S' are the focii.Can someone tell me what it is and how you use it? Its in ellipses.
I believe there are questions that require you to prove ratios of lengths initially and using the sine rule, to prove the reflection propertyIf P is a point on an ellipse, then the normal to the ellipse at P bisects SPS', where S and S' are the focii.
So if you and I were standing at the focii of a room with an ellipsoidal roof, then you would heard things I said really loudly.
Its the elliptical analogue of a parabola reflecting straight lines parallel to its axis to straight lines passing through the focus.
You should try to prove this .
The uses of this property are abundant, you should definitely understand it.
So when can you determine when to use it because from the definition it doesn't sound too useful... What information can it give you?If P is a point on an ellipse, then the normal to the ellipse at P bisects SPS', where S and S' are the focii.
So if you and I were standing at the focii of a room with an ellipsoidal roof, then you would heard things I said really loudly.
Its the elliptical analogue of a parabola reflecting straight lines parallel to its axis to straight lines passing through the focus.
You should try to prove this .
The uses of this property are abundant, you should definitely understand it.
How does it not sound useful? Reflection properties are literally the most practically applicable property of conics.So when can you determine when to use it because from the definition it doesn't sound too useful... What information can it give you?
You can also prove it using the triangle inequality - ala 2000 HSC Q7bMake sure you are aware of this property (as you obviously are) - it popped up in the 2011 HSC without an explanation for what it actually is. The BOS were allowed to do this because it is explicitly mentioned in the syllabus.
It can be proved myriad ways:
1. Using sine rule and then ratios of adjacent sides of triangles (the syllabus explicitly states that this style of proof is valid for HSC 4U)
2. Using perp distance formula from focus to tangent and using focus-directrix definition. This proof can be found in Terry Lee's book (as can the above proof)
3. Using the angle between two lines formula (tan theta = blah blah blah) - this is tedious.
My preferred method is the 2nd.
https://www.math.hmc.edu/funfacts/ffiles/10001.2-3.shtmlAlso how does this apply to hyperbola's ?
It depends. Firstly, I'm not sure how much heat from a fire is transmitted by radiation and how much is transmitted by convection. My suspicion is that at any reasonable distance from the fire, most heat comes from convection. Only radiated heat reflects (it is infrared radiation) - convected heat doesn't care too much for reflection laws.Dont you think the fire is kind of bs ??
But wouldn't it be illogical to say that standing at focus would be hotter than standing closer than the distance from the fire to the focus? So if i stand right next to the fire vs standing at the focusIt depends. Firstly, I'm not sure how much heat from a fire is transmitted by radiation and how much is transmitted by convection. My suspicion is that at any reasonable distance from the fire, most heat comes from convection. Only radiated heat reflects (it is infrared radiation) - convected heat doesn't care too much for reflection laws.
The second issue is the properties of the mirror. Most mirrors are designed only to reflect visible light, and any reflection of longer or shorter wavelengths would be only incidental.
Interestingly, I found a reference to what are called "hot mirrors" and "cold mirrors".
The cold mirrors I saw are designed to reflect 90% of visible light (so a normal mirror in that respect), while transmitting 80% of infrared radiation.
But the hot mirrors are the exact opposite. They reflect 90% of infrared and transmit 80% of visible light. So visibly, it doesn't act like a mirror, but it will reflect any heat you radiate on it.
If we assume the air cannot convect to you (an absurd assumption, but play along for now), then the only heat from the fire is radiation. Assuming we use perfectly reflective infrared mirrors (again, absurd), then (assuming we are in a mathematical universe), the radiation will perfectly converge onto the focus, heating the air to about the same temperature as the fire itself is (in terms of infrared heat). Convection wise, the fire is still hotter than the unoccupied focus.But wouldn't it be illogical to say that standing at focus would be hotter than standing closer than the distance from the fire to the focus? So if i stand right next to the fire vs standing at the focus