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General Equation of Hyperbola

Find the equa...

Question

Find the equaion of the hyperbola whose

(i) focus is (0,3), directrix is x+y-1=0 and eccentricity=2

(ii) foucs is (1,1), directrix is 3x+4y+8=0 and eccentricity=2

(iii) focus is (1,1), directrix is 2x+y=1 and 3ccentricity= $\sqrt{3}$

(iv) focus is 92,-1), firectrix is 2x+3y=1 and eccentricity =2.

(v) focus is (a,0).directrix is 2x-y+a=0 and eccentricity = $\frac{4}{3}$

(vi) focus is (2,2), directiex is x+y=9 and eccentricity =2.

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Solution

$(i) L e t S (0, 3) b e t h e f o c u s a n d P (x, y) b e a p o i n t o n t h e h y p e r b o l a .$

$Draw PM perpendicular from P on the directrix.Then by definition S P = e P M \Rightarrow S P^{2} = e P M^{2} \Rightarrow (x - 0)^{2}) + (y - 3)^{2} = (2)^{2} {[\begin{matrix} \frac{x - y + 1}{\sqrt{1^{2} + 1^{2}}} \end{matrix}]}^{2} [∵= 2] \Rightarrow x^{2} + y^{2} + 9 - 6 y = \frac{4 [x + y - 1]^{2}}{2} \Rightarrow x + 2 + y^{2} - 6 y + 9 = 2 (x + y - 1)^{2} \Rightarrow x + 2 + y^{2} - 6 y + 9 = 2 [x^{2} + y^{2} + (- 1)^{2} + 2 x y + 2 \times y \times (- 1) + 2 \times (- 1) \times x] \Rightarrow x^{2} + y^{2} - 6 y + 9 = 2 [x^{2} + y^{2} + 1 + 2 x y - 2 y - 2 x] \Rightarrow x^{2} + y^{2} - 6 y + 9 = 2 x^{2} + 2 y^{2} + 2 + 4 x y - 4 y - 4 x \Rightarrow 2 x^{2} - x^{2} + 2 y^{2} - y^{2} + 4 x y - 4 x - 4 y + 6 y + 2 - 9 = 0 \Rightarrow x^{2} + y^{2} + 4 x y - 4 x + 2 y - 7 = 0 This is the required equation of the hyperbola. (i i) Let S(1,1) be the focus and P(x,y)be a point on the hyperbola.$

$Draw PM perpendicular from P on the directrix .Then by definition S P = e P M \Rightarrow S P^{2} = e P M^{2} \Rightarrow (x - 1)^{2}) + (y - 1)^{2} = (2)^{2} {[\begin{matrix} \frac{3 x + 4 y + 8}{\sqrt{3^{2} + 4^{2}}} \end{matrix}]}^{2} [∵= 2] \Rightarrow x^{2} + 1 - 2 x + y^{2} + 1 - 2 y = 4 [\begin{matrix} \frac{3 x + 4 y + 8}{\sqrt{2} 5} \end{matrix}] \Rightarrow x^{2} + y^{2} - 2 x - 2 y + 2 = \frac{4 (2 x + 4 y + 8)^{2}}{25} \Rightarrow 23 x^{2} + 25 y^{2} - 50 x - 50 y + 50 = 4 [9 x^{2} + 16 y^{2} + 6 y + 24 x y + 64 y + 48 x] \Rightarrow 25 x^{2} + 25 y^{2} - 50 x - 50 y + 50 = 36 x^{2} + 64 y^{2} 56 + 96 x y + 256 y + 192 x \Rightarrow 36 x^{2} - 25 x^{2} + 64 y^{2} - 25 y^{2} + 96 x y + 192 x + 50 x + 256 y + 50 y + 256 - 50 = 0 \Rightarrow 11 x^{2} + 39 y^{2} + 96 x y + 242 x + 306 y + 206 = 0 This is the required equation of the hyperbola . (i i i) L e t S (1, 1) b e t h e f o c u s a n d P (x, y) b e a p o i n t o n t h e h y p e r b o l a .$

$Draw PM perpendicular from P on the directrix.Then by definition S P = e P M \Rightarrow S P^{2} = e^{2} P M^{2} \Rightarrow (x - 1)^{2}) + (y - 1)^{2} = (\sqrt{3})^{2} {[\begin{matrix} \frac{2 x - y - 1}{\sqrt{2^{2} + 2^{2}}} \end{matrix}]}^{2} [∵= 2] \Rightarrow x^{2} + 1 - 2 x + y^{2} + 1 - 2 y = \frac{3 [2 x - y - 1]^{2}}{5} \Rightarrow 5 [x^{2} + y^{2} - 2 x - 2 y + 2] = 3 (2 x + y - 1)^{2} \Rightarrow 5 x^{2} + 5 y^{2} - 10 x - 10 y + 10 = 3 [(2 x)^{2} + y^{2} + (- 1)^{2} + 2 \times 2 x \times y + 2 \times y \times (- 1) + 2 \times (- 1) \times 2 x] \Rightarrow 5 x^{2} + 5 y^{2} - 10 x - 10 y + 10 = 3 [4 x^{2} + y^{2} + 1 + 4 x y - 2 y - 4 x] \Rightarrow 5 x^{2} + 5 y^{2} - 10 x - 10 y + 10 = 12 x^{2} + 3 y^{2} + 3 + 12 x y - 6 y - 12 x \Rightarrow 12 x^{2} - 5 x^{2} + 3 y^{2} - 5 y^{2} + 12 x y - 12 x + 10 x - 6 y + 10 y + 3 - 10 = 0 \Rightarrow 7 x^{2} - 2 y^{2} + 12 x y - 2 x + 4 y - 7 = 0 This is the required equation of the hyperbola. (i v) L e t S (2, 1) b e t h e f o c u s a n d P (x, y) b e a p o i n t o n t h e h y p e r b o l a .$

$Draw PM perpendicular from P on the directrix.Then by definition S P = e P M \Rightarrow S P^{2} = e^{2} P M^{2} \Rightarrow (x - 2)^{2}) + (y + 1)^{2} = 2^{2} {[\begin{matrix} \frac{2 x + 3 y - 1}{\sqrt{2^{2} + 3^{2}}} \end{matrix}]}^{2} [∵ e = 2] \Rightarrow x^{2} + 4 - 4 x + y^{2} + 1 - 2 y = \frac{4 [2 x - 3 y - 1]^{2}}{13} \Rightarrow 13 [x^{2} + y^{2} - 4 x + 2 y + 5] = 4 (2 x + 3 y - 1)^{2} \Rightarrow 13 x^{2} + 13 y^{2} - 52 x + 26 y + 65 = 4 [2 x + 3 y - 1]^{2} \Rightarrow 13 x^{2} + 13 y^{2} - 52 x + 26 y + 65 = 4 \Rightarrow [(2 x^{2})^{2} + (3 y)^{2} + (- 1)^{2} + 2 \times 2 x \times 3 y + 2 \times 3 y \times (- 1) + 2 \times (- 1) \times 2 x] \Rightarrow 13 x^{2} + 13 y^{2} - 52 x + 26 y + 65 = 4 [4 x^{2} + 9 y^{2} + 1 + 12 x y - 6 y - 4 x] \Rightarrow 13 x^{2} + 13 y^{2} - 52 x + 26 y + 65 = 16 x^{2} + 36 y^{2} + 4 + 48 x y - 24 y - 16 x \Rightarrow 16 x^{2} - 13 x^{2} + 36 y^{2} + - 13 y^{2} + 48 x y - 16 x + 52 x - 24 y - 26 y + 4 - 65 = 0 \Rightarrow 3 x^{2} + 23 y^{2} + 48 x y + 36 x - 50 y - 61 = 0 This is the required equation of the hyperbola (v) Let S(a,0)be the focus and P(x,y)be a point on the hyperbola.$

$Draw PM perpendicular from P on the directrix .Then by definition S P = e P M \Rightarrow S P^{2} = e P M^{2} \Rightarrow (x - a)^{2} + (y - 0)^{2} = {(\frac{4}{3})}^{2} {[\begin{matrix} \frac{2 x - y + a}{\sqrt{2^{2} + (- 1)^{2}}} \end{matrix}]}^{2} [\begin{matrix} ∵ e = \frac{4}{3} \end{matrix}] \Rightarrow x^{2} + a^{2} - 2 a x + y^{2} = \frac{16}{9} \times \frac{[2 x - y + a]^{2}}{5} \Rightarrow 45 [x^{2} + y^{2} - 2 a x + a^{2}] = 16 [2 x - y + a]^{2} \Rightarrow 45 x^{2} + 45 y^{2} - 90 a x + 45 a^{2} = 16 [(2 x)^{2} + (- y)^{2} + a^{2} + 2 x (- y) + 2 \times (- y) \times a + 2 \times a \times 2 x] \Rightarrow 45 x^{2} + 45 y^{2} - 90 a x + 45 a^{2} = 16 [4 x^{2} + y^{2} + a^{2} - 4 x y - 2 a y + 4 a x] \Rightarrow 45 x^{2} + 45 y^{2} - 90 x + 45 a^{2} \Rightarrow 64 x^{2} + 16 y^{2} + 16 a^{2} - 64 x y - 32 a y + 64 a x \Rightarrow 64 x^{2} - 45 x^{2} + 16 y^{2} - 45 y^{2} - 64 x y + 64 a x + 90 a x - 32 a y + 16 a^{2} - 45 a^{2} = 0 \Rightarrow 19 x^{2} - 29 y^{2} - 64 x y + 154 a x - 32 a y - 29 a^{2} = 0 This is the required equation of the hyperbola. (v i) Let S (2,2)be the focus and P(x,y)be a point on the hyperbola$

$Draw PM perpendicular from P on the directrix.Then by definition S P = e P M \Rightarrow S P^{2} = e^{2} P M^{2} \Rightarrow (x - 2)^{2} + (y - 2)^{2} = 2^{2} {[\begin{matrix} \frac{x + y - 9}{\sqrt{1^{2} + 1^{2}}} \end{matrix}]}^{2} [\begin{matrix} ∵ e = \frac{4}{3} \end{matrix}] \Rightarrow x^{2} + 4 - 4 x + y^{2} + 4 - 4 y = \frac{4 [x + y - 9]^{2}}{2} \Rightarrow x^{2} + y^{2} - 4 x - 4 y + 8 = 2 [x + y - 9]^{2} \Rightarrow x^{2} + y^{2} - 4 x - 4 y + 8 = 2 [x^{2} + y^{2} + (- 9)^{2} + 2 \times x \times y + 2 \times y \times (- 9) + 2 \times (- 9) \times x] \Rightarrow x^{2} + y^{2} - 4 x - 4 y + 8 = 2 [x^{2} + y^{2} + 81 + 2 x y - 18 y + 18 x] \Rightarrow x^{2} + y^{2} - 4 x - 4 y + 8 = [2 x^{2} + 2 y^{2} + 162 + 4 x y + 4 x y - 36 y - 36 x] \Rightarrow 2 x^{2} - x^{2} + 2 y^{2} - y^{2} + 4 x y - 36 x + 4 x - 36 y + 4 y + 162 - 8 = 0 \Rightarrow x^{2} + y^{2} + 4 x y - 32 x - 32 y + 154 = 0 This is the required equation of the hyperbola.$

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Similar questions

Q. Find the equation of the hyperbola whose
(i) focus is (0, 3), directrix is x + y − 1 = 0 and eccentricity = 2
(ii) focus is (1, 1), directrix is 3x + 4y + 8 = 0 and eccentricity = 2
(iii) focus is (1, 1) directrix is 2x + y = 1 and eccentricity =

\sqrt{3}

(iv) focus is (2, −1), directrix is 2x + 3y = 1 and eccentricity = 2
(v) focus is (a, 0), directrix is 2x − y + a = 0 and eccentricity =

\frac{4}{3}

(vi) focus is (2, 2), directrix is x + y = 9 and eccentricity = 2.

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