Rectangular Hyperbola Conics HSC Maths Extension 2

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Randolph Rice
5 years ago
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1 Rectangular Hyperbola HSC Maths Etension

2 Question For the y, find: (c) (d) (e) the eccentricity the coordinates of the foci the equations of the directrices the equations of the asymptotes sketch the hyperbola. Question Find the equation of the normal to the y at the point,6. Hence, find where the normal meets the curve again. Question Find the coordinates of the foci and the equations of the directrices of the e y 0 y 8 (c) y 4 (d) y Question 4 Show that there are generally four normals that can be drawn from a point to the hyperbola y c. Find the equations of the normals drawn to the hyperbola y 4 from the point 5,5.

3 Rectangular Hyperbola Question 5 The chord PQ of the AP BQ y c meets the aes in A and B. Prove that The chord PQ of the y c meets the -ais in B. If O is the centre and R is the mid point of PQ, show that OR RB. Question 6 A variable chord of the y c passes through the fied point, Prove that the locus of the midpoint of the chord is given by y y.. Question 7 Find the condition for p, q if the chord joining P cp, c p, Q cq, c q of the y c. passes through a focus Find the locus of the midpoint of PQ if PQ is a focal chord.

4 Fully Worked Solutions Question a b b a e e S c, c, Sc, c c 4 S 8,8, S8, 8 (c) y c, thus y 4 y 8 (d) are 0, y 0 (e) Question y, y for, m, therefore m 4 the equation of the normal is y 6 y 6 0 Solve simultaneously y6 0 and y Question y6 y y y 6 0 y or 6 Put y in y 6 0 gives 8 therefore the normal meets the curve again in y 0, ab 0, y 8, ab 8, (c) y 4, c 4, c, 8, e, foci 0,0 e, foci 6,0, directrices 5, directrices e, foci c, c,,

5 4 Rectangular Hyperbola directrices y c (d) y, c, c, Question 4 c cp, y p c dy dy dp p d d / dp c p / e, Foci c, c 4, 4 The equation of the normal y p cp 4 py c p cp 4 p py c p If this normal goes through c p,,, p p c p 4 4 cp p p c 0 This is a quartic in terms of p, so, there generally are four values of p that satisfy this equation. As each value of p corresponds to a point, there generally are four normals to the hyperbola through a given point. 4 4, in cp p p c 0 5,5 and c, p 5p 5p 0 Substituting by By inspection, we find that p and are two roots. 4 p 5p 5p p p p 5p 0 therefore p,,,. Corresponding to these values of p, there are three normals: Put p in, noting c, gives y 0 Put p in () gives 4y5 0 Put p in () gives 4 y5 0. () Question 5 Let P cp, c p, Q cq, c q c c p q cq p mpq cp cq c p q pq pq The equation of the chord PQ is: y cp pqy c p q 0 c p pq. pqy cq cp The chord meets the aes in A 0, c p q and Bc p q,0 pq To prove that AP BQ we ll prove that AB and PQ have the same midpoint.

6 5 c p q c p q The midpoint of AB is, pq cp cq c c c p q c p q The midpoint of PQ is R,, p q pq which is the same as the midpoint of AB. therefore AP BQ The gradient of OR is c p q m c p q pq pq The gradient of PQ is m pq Since m m, ROB RBO therefore ROB is isosceles, therefore OR RB Question 6 the equation of the chord PQ is pqy c p q 0 The chord passes through,, so pq c p q 0 () c p q c c c p q But the coordinates of the midpoint of PQ are, y p q pq therefore pq c and c p q pq y y y Put to (), c 0 y c y y 0 therefore the locus of the midpoint of PQ is y y 0 Question 7 The equation of the chord PQ is pqy c p q 0 The chord passes through c, c, so c pq c c p q 0 pq p q 0 p q therefore pq c p q c c c p q But the coordinates of the midpoint of PQ are, y p q pq therefore pq c and c p q pq y y y p q Put to, c y pq c y y

7 6 Rectangular Hyperbola therefore the locus of the midpoint of PQ is y c y y c y

8 7

Equations of Ellipse Conics HSC Maths Extension 2

Equations of Ellipse Conics HSC Maths Extension 2 Equations of Ellipse HSC Maths Extension 1 Question 1 Find the equation of the ellipse whose foci on the y-axis, centre 0,0, a, b. Question Find the equation of the ellipse whose foci 4,0, b. Question