Question

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.

Answer 1

(i) Let S be the focus and $P\left(x,y\right)$ be any point on the hyperbola.
Draw PM perpendicular to the directrix.

By definition:
SP = ePM
$\Rightarrow$ $\sqrt{(x-0{)}^2+(y-3{)}^2}=2\left(\frac{x+y-1}{\sqrt{2}}\right)$
Squaring both the sides:
$$\begin{gathered} (x-0{)}^2+(y-3{)}^2=4{\left(\frac{x+y-1}{\sqrt{2}}\right)}^2 \\ \Rightarrow x^2+y^2+9-6y=2\left(x^2+y^2+1+2xy-2y-2x\right) \\ \Rightarrow x^2+y^2+4xy+2y-4x-7=0 \end{gathered}$$
∴ Equation of the hyperbola = $x^2+y^2+4xy+2y-4x-7=0$

(ii) Let S be the focus and $P\left(x,y\right)$ be any point on the hyperbola.
Draw PM perpendicular to the directrix.
By definition:
SP = ePM


$\Rightarrow$$\sqrt{(x-1{)}^2+(y-1{)}^2}=2\left(\frac{3x+4y+8}{5}\right)$
Squaring both the sides:
$$\begin{gathered} (x-1{)}^2+(y-1{)}^2=4{\left(\frac{3x+4y+8}{5}\right)}^2 \\ \Rightarrow x^2+1-2x+y^2+1-2y=\frac{4}{25}\left(9x^2+16y^2+64+24xy+64y+48x\right) \\ \Rightarrow 25x^2+25-50x+25y^2+25-50y=36x^2+64y^2+256+96xy+256y+192x \\ \Rightarrow 11x^2+39y^2+96xy+306y+242x+206=0 \end{gathered}$$
∴ Equation of the hyperbola = $11x^2+39y^2+96xy+306y+242x+206=0$

(iii) Let S be the focus and $P\left(x,y\right)$ be any point on the hyperbola.
Draw PM perpendicular to the directrix.
By definition:
SP = ePM

$\Rightarrow$$\sqrt{(x-1{)}^2+(y-1{)}^2}=\sqrt{3}\left(\frac{2x+y-1}{\sqrt{5}}\right)$
Squaring both the sides:
$$\begin{gathered} (x-1{)}^2+(y-1{)}^2=3{\left(\frac{2x+y-1}{5}\right)}^2 \\ \Rightarrow x^2+1-2x+y^2+1-2y=\frac{3}{5}\left(4x^2+y^2+1+4xy-2y-4x\right) \\ \Rightarrow 5x^2+5-10x+5y^2+5-10y=12x^2+3y^2+3+12xy-6y-12x \\ \Rightarrow 7x^2-2y^2+12xy+4y-2x-7=0 \end{gathered}$$
∴ Equation of the hyperbola = $7x^2-2y^2+12xy+4y-2x-7=0$

(iv) Let S be the focus and $P\left(x,y\right)$ be any point on the hyperbola.
Draw PM perpendicular to the directrix.




By definition:
SP = ePM
= ePM
$\Rightarrow$$\sqrt{(x-2{)}^2+(y+1{)}^2}=2\left(\frac{2x+3y-1}{\sqrt{13}}\right)$
Squaring both the sides:
$$\begin{gathered} (x-2{)}^2+(y+1{)}^2=4{\left(\frac{2x+3y-1}{13}\right)}^2 \\ \Rightarrow x^2+4-4x+y^2+1+2y=\frac{4}{13}\left(4x^2+9y^2+1+12xy-6y-4x\right) \\ \Rightarrow 13x^2+52-52x+13y^2+13+26y=16x^2+36y^2+4+48xy-24y-16x \\ \Rightarrow 3x^2+23y^2+48xy-50y+36x-61=0 \end{gathered}$$
∴ Equation of the hyperbola = $3x^2+23y^2+48xy-50y+36x-61=0$

(v) Let S be the focus and $P\left(x,y\right)$ be any point on the hyperbola.
Draw PM perpendicular to the directrix.


By definition:
SP = ePM
$\Rightarrow$$\sqrt{(x-a{)}^2+(y-0{)}^2}=\frac{4}{3}\left(\frac{2x-y+a}{\sqrt{5}}\right)$
Squaring both the sides:
$$\begin{gathered} (x-a{)}^2+(y{)}^2=\frac{16}{9}{\left(\frac{2x-y+a}{5}\right)}^2 \\ \Rightarrow x^2-2ax+a^2+y^2=\frac{16}{45}\left(4x^2+y^2+a^2-4xy-2ya+4xa\right) \\ \Rightarrow 45x^2-90ax+45a^2+45y^2=64x^2+16y^2+16a^2-64xy-32ay+64ax \\ \Rightarrow 19x^2-29y^2-64xy-32ay+154ax-29a^2=0 \end{gathered}$$
∴ Equation of the hyperbola = $19x^2-29y^2-64xy-32ay+154ax-29a^2=0$

(vi) Let S be the focus and $P\left(x,y\right)$ be any point on the hyperbola.
Draw PM perpendicular to the directrix.
By definition:
SP = ePM

$\Rightarrow$$\sqrt{(x-2{)}^2+(y-2{)}^2}=2\left(\frac{x+y-9}{\sqrt{2}}\right)$
Squaring both the sides:
$$\begin{gathered} (x-2{)}^2+(y-2{)}^2=4{\left(\frac{x+y-9}{2}\right)}^2 \\ \Rightarrow x^2-4x+4+y^2-4y+4=2\left(x^2+y^2+81+2xy-18y-18x\right) \\ \Rightarrow x^2-4x+4+y^2-4y+4=2x^2+2y^2+162+4xy-36y-36x \\ \Rightarrow x^2+y^2+4xy-32y-32x+154=0 \end{gathered}$$
∴ Equation of the hyperbola = $x^2+y^2+4xy-32y-32x+154=0$