Before I post any questions, I would like to make a couple of introductory comments about conics.
Most students don't like conics, saying it is hard. I disagree. This is (conceptual) a relatively straight-forward topic, based mostly in simple co-ordinate geometry and calculus. What makes it difficult is that the algebra is often tangled and lengthy.
So, IMO, the key to conics is to minimise (to the greatest extent possible) the algebra. Easier said than done, you might be thinking. Well, here are some suggestions.
1. Draw a diagram. Make it big. Make sure you have plenty of space to draw onto it, should that be needed.
2. Think about the question. Most questions have an obvious algebra-bash approach. You are trying to find an alternative, because there usually is one.
3. Alternatives:
(a) Look to use properties of the conic sections, like the focus-directrix definition PS = ePM.
(b) Look for recognisable geometry shapes, triangles, etc.
(c) Look for connections between those shapes - similar triangles, congruent triangles, etc
(d) Look to make diagonals into horizontals and verticals - if you have to find a distance, you want it to be horizontal or vertical so that you can avoid the distance formula.
(e) Look for other results, like intercept theorems, or trigonometry.
4. Only after you have exhausted the above may you consider an algebra-bash.
Now, here are some questions to think about - I'm looking for two types of solutions - an elegant solution (if you can find one), or an algebra bash if you can't. Have fun!
1. The point P is any point on the ellipse x<sup>2</sup> / a<sup>2</sup> + y<sup>2</sup> / b<sup>2</sup> = 1, which has foci at S and S'. Prove that PS + PS' = 2a. (Note: this is a standard problem. Anyone who does not know a really fast way to do this should learn it as soon as one of your colleagues posts it.)
2. The point P(acos@, bsin@) is a point on the ellipse x<sup>2</sup> / a<sup>2</sup> + y<sup>2</sup> / b<sup>2</sup> = 1. The tangent at P meets the x-axis at T, and the normal at P meets the x-axis at N. Draw a diagram to represent this information, and show that
(PT / PN)<sup>2</sup> = tan<sup>2</sup>@ / (1 - e<sup>2</sup>).
3(a). ABC is a triangle and X is a point on BC. Prove that AX bisects angle BAC if and only if AB:AC = BX:CX, using the sine rule or otherwise.
(b) P(asec@, btan@) is any point on the hyperbola x<sup>2</sup> / a<sup>2</sup> - y<sup>2</sup> / b<sup>2</sup> = 1, which is not located on the x-axis. The tangent at P meets the x-axis at T, and S and S' are the foci of the hyperbola.
(i) Show that T has coordinates (acos@, 0)
(ii) Prove that PS and PS' are inclined to the tangent at P at equal angles.
4. P(acos@, bsin@) and Q(acos#, bsin#) are two distinct points on the ellipse x<sup>2</sup> / a<sup>2</sup> + y<sup>2</sup> / b<sup>2</sup> = 1, and the chord PQ subtends a right angle at A(a, 0). Find an expression for tan(@ / 2) * tan(# / 2) in terms of a and b only.
5. P(asec@, btan@) is a point on the hyperbola x<sup>2</sup> / a<sup>2</sup> - y<sup>2</sup> / b<sup>2</sup> = 1, and tan@ <> 0. The tangent at P meets the x- and y- axes at X and Y, respectively, and meets the asymptotes at K and L.
(a) Find the equation of the tangent at P, and hence find the coordinates of X and Y.
(b) Find the value of
(i) PX / XY, and
(ii) PX / PY
(c) Find the value of PK / PL
Most students don't like conics, saying it is hard. I disagree. This is (conceptual) a relatively straight-forward topic, based mostly in simple co-ordinate geometry and calculus. What makes it difficult is that the algebra is often tangled and lengthy.
So, IMO, the key to conics is to minimise (to the greatest extent possible) the algebra. Easier said than done, you might be thinking. Well, here are some suggestions.
1. Draw a diagram. Make it big. Make sure you have plenty of space to draw onto it, should that be needed.
2. Think about the question. Most questions have an obvious algebra-bash approach. You are trying to find an alternative, because there usually is one.
3. Alternatives:
(a) Look to use properties of the conic sections, like the focus-directrix definition PS = ePM.
(b) Look for recognisable geometry shapes, triangles, etc.
(c) Look for connections between those shapes - similar triangles, congruent triangles, etc
(d) Look to make diagonals into horizontals and verticals - if you have to find a distance, you want it to be horizontal or vertical so that you can avoid the distance formula.
(e) Look for other results, like intercept theorems, or trigonometry.
4. Only after you have exhausted the above may you consider an algebra-bash.
Now, here are some questions to think about - I'm looking for two types of solutions - an elegant solution (if you can find one), or an algebra bash if you can't. Have fun!
1. The point P is any point on the ellipse x<sup>2</sup> / a<sup>2</sup> + y<sup>2</sup> / b<sup>2</sup> = 1, which has foci at S and S'. Prove that PS + PS' = 2a. (Note: this is a standard problem. Anyone who does not know a really fast way to do this should learn it as soon as one of your colleagues posts it.)
2. The point P(acos@, bsin@) is a point on the ellipse x<sup>2</sup> / a<sup>2</sup> + y<sup>2</sup> / b<sup>2</sup> = 1. The tangent at P meets the x-axis at T, and the normal at P meets the x-axis at N. Draw a diagram to represent this information, and show that
(PT / PN)<sup>2</sup> = tan<sup>2</sup>@ / (1 - e<sup>2</sup>).
3(a). ABC is a triangle and X is a point on BC. Prove that AX bisects angle BAC if and only if AB:AC = BX:CX, using the sine rule or otherwise.
(b) P(asec@, btan@) is any point on the hyperbola x<sup>2</sup> / a<sup>2</sup> - y<sup>2</sup> / b<sup>2</sup> = 1, which is not located on the x-axis. The tangent at P meets the x-axis at T, and S and S' are the foci of the hyperbola.
(i) Show that T has coordinates (acos@, 0)
(ii) Prove that PS and PS' are inclined to the tangent at P at equal angles.
4. P(acos@, bsin@) and Q(acos#, bsin#) are two distinct points on the ellipse x<sup>2</sup> / a<sup>2</sup> + y<sup>2</sup> / b<sup>2</sup> = 1, and the chord PQ subtends a right angle at A(a, 0). Find an expression for tan(@ / 2) * tan(# / 2) in terms of a and b only.
5. P(asec@, btan@) is a point on the hyperbola x<sup>2</sup> / a<sup>2</sup> - y<sup>2</sup> / b<sup>2</sup> = 1, and tan@ <> 0. The tangent at P meets the x- and y- axes at X and Y, respectively, and meets the asymptotes at K and L.
(a) Find the equation of the tangent at P, and hence find the coordinates of X and Y.
(b) Find the value of
(i) PX / XY, and
(ii) PX / PY
(c) Find the value of PK / PL
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