Given Equation: \[\frac{{{x}^{2}}}{25}-\frac{{{y}^{2}}}{4}=1\] Comparing with the equation of hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] we get, a = 5 and b = 2 (v)...

### Find the (v) length of the rectum of each of the following the hyperbola :

### Find the equation of the hyperbola, referred to its principal axes as axes of coordinates, in the following cases: conjugate axis is 7 and passes through the point (3, -2)

conjugate axis is \[7\]and passes through the point \[\left( 3,\text{ }-2 \right)\] Given: Conjugate axis \[=\text{ }7\] Passes through the point \[\left( 3,\text{ }-2 \right)\] Conjugate axis is...

### Find the equation of the hyperbola, referred to its principal axes as axes of coordinates, in the following cases: (i) the distance between the foci = 16 and eccentricity = √2 (ii) conjugate axis is 5 and the distance between foci = 13

(i) the distance between the \[foci\text{ }=\text{ }16\text{ }and\text{ }eccentricity\text{ }=\text{ }\surd 2\] Given: Distance between the foci \[=\text{ }16\] Eccentricity \[=\text{ }\surd 2\] Let...

### Find the centre, eccentricity, foci and directions of the hyperbola x^2 – 3y^2 – 2x = 8

\[{{x}^{2}}-\text{ }3{{y}^{2}}-\text{ }2x\text{ }=\text{ }8\] Given: The equation \[=>\text{ }{{x}^{2}}-\text{ }3{{y}^{2}}-\text{ }2x\text{ }=\text{ }8\] Let us find the centre, eccentricity,...

### Find the centre, eccentricity, foci and directions of the hyperbola (i) 16x^2 – 9y^2 + 32x + 36y – 164 = 0 (ii) x^2 – y^2 + 4x = 0

(i) \[16{{x}^{2}}-\text{ }9{{y}^{2}}~+\text{ }32x\text{ }+\text{ }36y\text{ }-\text{ }164\text{ }=\text{ }0\] Given: The equation \[=>\text{ }16{{x}^{2}}-\text{ }9{{y}^{2}}~+\text{ }32x\text{...

### Find the axes, eccentricity, latus-rectum and the coordinates of the foci of the hyperbola 25x^2 – 36y^2 = 225.

Given: The equation\[=>\text{ }25{{x}^{2}}-\text{ }36{{y}^{2}}~=\text{ }225\] The equation can be expressed as: The obtained equation is of the form Where, \[a\text{ }=\text{ }3\text{ }and\text{...

### Find the eccentricity, coordinates of the foci, equations of directrices and length of the latus-rectum of the hyperbola. 2x^2 – 3y^2 = 5

\[2{{x}^{2}}-\text{ }3{{y}^{2}}~=\text{ }5\] Given: The equation \[=>\text{ }2{{x}^{2}}-\text{ }3{{y}^{2}}~=\text{ }5\] The equation can be expressed as: The obtained equation is of the form...

### Find the eccentricity, coordinates of the foci, equations of directrices and length of the latus-rectum of the hyperbola. (i) 4x^2 – 3y^2 = 36 (ii) 3x^2 – y^2 = 4

(i) \[4{{x}^{2}}-\text{ }3{{y}^{2}}~=\text{ }36\] Given: The equation \[=>\text{ }4{{x}^{2}}-\text{ }3{{y}^{2}}~=\text{ }36\] The equation can be expressed as: The obtained equation is of the...

### Find the eccentricity, coordinates of the foci, equations of directrices and length of the latus-rectum of the hyperbola. (i) 9×2 – 16y2 = 144 (ii) 16×2 – 9y2 = -144

(i) \[9{{x}^{2}}-\text{ }16{{y}^{2}}~=\text{ }144\] Given: The equation \[=>\text{ }9{{x}^{2}}-\text{ }16{{y}^{2}}~=\text{ }144\] The equation can be expressed as: The obtained equation is of the...

### Find the equation of the hyperbola whose (i) focus is (a, 0), directrix is 2x + 3y = 1 and eccentricity = 2 (ii)focus is (2, 2), directrix is x + y = 9 and eccentricity = 2

(i) \[focus\text{ }is\text{ }\left( a,\text{ }0 \right),\]\[directrix\text{ }is\text{ }2x\text{ }+\text{ }3y\text{ }=\text{ }1\]and \[eccentricity\text{ }=\text{ }2\] Given: \[Focus\text{ }=\text{...

### Find the equation of the hyperbola whose (i) focus is (1, 1) directrix is 2x + y = 1 and eccentricity =√3 (ii) focus is (2, -1), directrix is 2x + 3y = 1 and eccentricity = 2

(i) \[focus\text{ }is\text{ }\left( 1,\text{ }1 \right)\]\[directrix\text{ }is\text{ }2x\text{ }+\text{ }y\text{ }=\text{ }1\] and \[eccentricity\text{ }=\surd 3\] Given: \[Focus\text{ }=\text{...

### 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

(i) focus is \[\left( 0,\text{ }3 \right),\] directrix is \[x\text{ }+\text{ }y\text{ }-\text{ }1\text{ }=\text{ }0\] and eccentricity \[=\text{ }2\] Given: Focus \[=\text{ }\left( 0,\text{ }3...

### The equation of the directrix of a hyperbola is x – y + 3 = 0. Its focus is (-1, 1) and eccentricity 3. Find the equation of the hyperbola.

Given: The equation of the directrix of a hyperbola \[=>\text{ }x\text{ }-\text{ }y\text{ }+\text{ }3\text{ }=\text{ }0\] Focus \[=\text{ }\left( -1,\text{ }1 \right)\] and Eccentricity \[=\text{...