Let $H : \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1, a > 0, b > 0$, be a hyperbola such that the sum of lengths of the transverse and the conjugate axes is $4(2\sqrt{2} + \sqrt{14})$. If the eccentricity $H$ is $\frac{\sqrt{11}}{2}$, then value of $a^2 + b^2$ is equal to ______.

### Analyzing the Setup The hyperbola is defined by the standard equation: $$ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $$ The transverse axis has a length of $2a$, and the conjugate axis has a length of $2b$. We are given the constraint: $$ 2a + 2b = 4(2\sqrt{2} + \sqrt{14}) $$ ### The Bridge of Eccentricity The eccentricity $e$ is given as $e = \frac{\sqrt{11}}{2}$. We utilize the fundamental relationship for a hyperbola: $$ e^2 = 1 + \frac{b^2}{a^2} $$ Substituting $e^2 = \frac{11}{4}$ into the equation, we obtain: $$ \frac{11}{4} = 1 + \frac{b^2}{a^2} \Rightarrow \frac{b^2}{a^2} = \frac{7}{4} $$ Taking the square root of both sides, we express the conjugate axis in terms of the transverse axis: $$ b = \frac{\sqrt{7}}{2}a $$ ### The Algebraic Symphony Returning to our initial constraint $a + b = 2(2\sqrt{2} + \sqrt{14})$, we substitute $b = \frac{\sqrt{7}}{2}a$: $$ a + \frac{\sqrt{7}}{2}a = 4\sqrt{2} + 2\sqrt{14} $$ Factoring $a$ on the left and simplifying the right side by factoring out $2\sqrt{2}$: $$ a\left(\frac{2 + \sqrt{7}}{2}\right) = 2\sqrt{2}(2 + \sqrt{7}) $$ Canceling the common term $(2 + \sqrt{7})$ from both sides, we find: $$ \frac{a}{2} = 2\sqrt{2} \Rightarrow a = 4\sqrt{2} $$ ### Final Calculation Now, we calculate the squares of the semi-axes. Squaring $a$ gives: $$ a^2 = (4\sqrt{2})^2 = 16 \times 2 = 32 $$ Using the ratio $\frac{b^2}{a^2} = \frac{7}{4}$, we determine $b^2$: $$ b^2 = \frac{7}{4} \times 32 = 7 \times 8 = 56 $$ The final sum is: $$ a^2 + b^2 = 32 + 56 = \mathbf{88} $$

Why did we factor out $2\sqrt{2}$ in the final steps?

Algebra is often about finding patterns. When we had the equation $a(1 + \frac{\sqrt{7}}{2}) = 4\sqrt{2} + 2\sqrt{14}$, we noticed that $2\sqrt{14}$ is actually $2\sqrt{2} \cdot \sqrt{7}$. Factoring out $2\sqrt{2}$ revealed the common term $(2 + \sqrt{7})$ on both sides, which allowed us to simplify the equation elegantly. Always look for common factors when dealing with radical expressions!

JEE Main 2022 (29 June Shift 1)

LEVELJEE Main

Animated Solution for Mathematics - Conic Sections: Let $H : \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1, a > 0, b > 0$ , be a hyperbola such that the sum of lengths of the transverse and the conjugate axes is $4 (22 + 14)$ . If the eccentricity $H$ is $\frac{11}{2}$ , then value of $a^{2} + b^{2}$ is equal to ______.

Enter Numerical Value:

Visualized Solution

The Hyperbola and its Axes

Standard hyperbola: $\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$
Transverse axis length $= 2 a$
Conjugate axis length $= 2 b$

Sum of the Axes

Given sum of axes: $2 a + 2 b = 4 (22 + 14)$

Simplifying the Sum Equation

Divide by $2$ : $a + b = 2 (22 + 14)$
Expand: $a + b = 42 + 214$

The Eccentricity Formula

Eccentricity formula: $e^{2} = 1 + \frac{b ^{2}}{a ^{2}}$
Given eccentricity: $e = \frac{11}{2}$

Squaring the Eccentricity

Square the given value: $e^{2} = (\frac{11}{2})^{2} = \frac{11}{4}$
Substitute into formula: $\frac{11}{4} = 1 + \frac{b ^{2}}{a ^{2}}$

Finding the Ratio $\frac{b ^{2}}{a ^{2}}$

Rearrange: $\frac{b ^{2}}{a ^{2}} = \frac{11}{4} - 1$
Simplify: $\frac{b ^{2}}{a ^{2}} = \frac{11 - 4}{4} = \frac{7}{4}$

Expressing $b$ in terms of $a$

Take square root: $\frac{b}{a} = \frac{7}{4} = \frac{7}{2}$
Cross-multiply: $b = \frac{7}{2} a$

Substitution into the Sum Equation

Recall: $a + b = 42 + 214$
Substitute $b$ : $a + \frac{7}{2} a = 42 + 214$

Factoring and Common Denominator

Factor out $a$ : $a (1 + \frac{7}{2}) = 42 + 214$
Common denominator: $a (\frac{2 + 7}{2}) = 42 + 214$

Solving for $a$

Factor right side: $22 (2 + 7)$
Equation: $a (\frac{2 + 7}{2}) = 22 (2 + 7)$
Cancel $(2 + 7)$ and solve: $a = 42$

Calculating $a^{2}$

We have $a = 42$
Square both sides: $a^{2} = (42)^{2}$
Calculate: $a^{2} = 16 \times 2 = 32$

Calculating $b^{2}$

Recall ratio: $b^{2} = \frac{7}{4} a^{2}$
Substitute $a^{2} = 32$ : $b^{2} = \frac{7}{4} (32)$
Calculate: $b^{2} = 7 \times 8 = 56$

Final Result: $a^{2} + b^{2}$

Calculate sum: $a^{2} + b^{2} = 32 + 56$
Final Answer: $88$

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The Sigma Insight: Foci, Directrices, and Eccentricity

Solution Diagram

Analyzing the Setup

The hyperbola is defined by the standard equation:

\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1

The transverse axis has a length of

2 a

, and the conjugate axis has a length of

2 b

. We are given the constraint:

2 a + 2 b = 4 (22 + 14)

The Bridge of Eccentricity

The eccentricity

e

is given as

e = \frac{11}{2}

. We utilize the fundamental relationship for a hyperbola:

e^{2} = 1 + \frac{b ^{2}}{a ^{2}}

Substituting

e^{2} = \frac{11}{4}

into the equation, we obtain:

\frac{11}{4} = 1 + \frac{b ^{2}}{a ^{2}} \Rightarrow \frac{b ^{2}}{a ^{2}} = \frac{7}{4}

Taking the square root of both sides, we express the conjugate axis in terms of the transverse axis:

b = \frac{7}{2} a

The Algebraic Symphony

Returning to our initial constraint

a + b = 2 (22 + 14)

, we substitute

b = \frac{7}{2} a

a + \frac{7}{2} a = 42 + 214

Factoring

a

on the left and simplifying the right side by factoring out

22

a (\frac{2 + 7}{2}) = 22 (2 + 7)

Canceling the common term

(2 + 7)

from both sides, we find:

\frac{a}{2} = 22 \Rightarrow a = 42

Final Calculation

Now, we calculate the squares of the semi-axes. Squaring

a

gives:

a^{2} = (42)^{2} = 16 \times 2 = 32

Using the ratio

\frac{b ^{2}}{a ^{2}} = \frac{7}{4}

, we determine

b^{2}

b^{2} = \frac{7}{4} \times 32 = 7 \times 8 = 56

The final sum is:

a^{2} + b^{2} = 32 + 56 = 88

Common Traps & FAQs

Wait, why is the transverse axis

2 a

and not just

a

That is a fantastic question! In the standard equation of a hyperbola,

\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1

, the vertices are at

(a, 0)

and

(- a, 0)

. The distance between these two vertices is the length of the transverse axis, which is

a - (- a) = 2 a

. It is a common trap to confuse the parameter

a

with the total length of the axis.

Can I use the relation

b^{2} = a^{2} (e^{2} - 1)

instead of

e^{2} = 1 + \frac{b ^{2}}{a ^{2}}

Absolutely! Both equations are algebraically equivalent. Whether you start with

e^{2} = 1 + \frac{b ^{2}}{a ^{2}}

or rearrange it to

b^{2} = a^{2} (e^{2} - 1)

, you are walking the same path. Use whichever feels more intuitive to you.

Why did we factor out

22

in the final steps?

Algebra is often about finding patterns. When we had the equation

a (1 + \frac{7}{2}) = 42 + 214

, we noticed that

214

is actually

22 \cdot 7

. Factoring out

22

revealed the common term

(2 + 7)

on both sides, which allowed us to simplify the equation elegantly. Always look for common factors when dealing with radical expressions!

Enter Numerical Value:

Visualized Solution

The Hyperbola and its Axes

Sum of the Axes

Simplifying the Sum Equation

The Eccentricity Formula

Squaring the Eccentricity

Finding the Ratio $\frac{b ^{2}}{a ^{2}}$

Expressing $b$ in terms of $a$

Substitution into the Sum Equation

Factoring and Common Denominator

Solving for $a$

Calculating $a^{2}$

Calculating $b^{2}$

Final Result: $a^{2} + b^{2}$

The Sigma Insight: Foci, Directrices, and Eccentricity

Analyzing the Setup

The Bridge of Eccentricity

The Algebraic Symphony

Final Calculation

Similar Questions

The eccentricity of the hyperbola whose length of the latus rectum is equal to 8 and the length of its conjugate axis is equal to half of the distance between its foci, is :

Let the product of the focal distances of the point $P (4, 23)$ on the hyperbola $H : \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$ be 32. Let the length of the conjugate axis of $H$ be $p$ and the length of its latus rectum be $q$ . Then $p^{2} + q^{2}$ is equal to _______

Let an ellipse with centre $(1, 0)$ and latus rectum of length $\frac{1}{2}$ have its major axis along x-axis. If its minor axis subtends an angle $6 0^{\circ}$ at the foci, then the square of the sum of the lengths of its minor and major axes is equal to _______.

If a hyperbola has length of its conjugate axis equal to 5 and the distance between its foci is 13, then the eccentricity of the hyperbola is:

If $e_{1}$ and $e_{2}$ are eccentricities of the ellipse $\frac{x ^{2}}{18} + \frac{y ^{2}}{4} = 1$ and the hyperbola $\frac{x ^{2}}{9} - \frac{y ^{2}}{4} = 1$ respectively and if the point $(e_{1}, e_{2})$ lies on ellipse $15 x^{2} + 3 y^{2} = k$ . Then find value of $k$

Let $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1 (a > b)$ be a given ellipse, length of whose latus rectum is 10. If its eccentricity is the maximum value of the function, $ϕ (t) = \frac{5}{12} + t - t^{2}$ , then $a^{2} + b^{2}$ is equal to :

Let one focus of the hyperbola $H : \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$ be at $(10, 0)$ and the corresponding directrix be $x = \frac{9}{10}$ . If $e$ and $l$ respectively are the eccentricity and the length of the latus rectum of $H$ , then $9 (e^{2} + l)$ is equal to:

List-I

List-II

Enter Numerical Value:

Visualized Solution

The Hyperbola and its Axes

Sum of the Axes

Simplifying the Sum Equation

The Eccentricity Formula

Squaring the Eccentricity

Finding the Ratio .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } a2b2​

Expressing .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } b in terms of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } a

Substitution into the Sum Equation

Factoring and Common Denominator

Solving for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } a

Calculating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } a2

Calculating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } b2

Final Result: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } a2+b2

The Sigma Insight: Foci, Directrices, and Eccentricity

Analyzing the Setup

The Bridge of Eccentricity

The Algebraic Symphony

Final Calculation

Similar Questions

The eccentricity of the hyperbola whose length of the latus rectum is equal to 8 and the length of its conjugate axis is equal to half of the distance between its foci, is :

If a hyperbola has length of its conjugate axis equal to 5 and the distance between its foci is 13, then the eccentricity of the hyperbola is:

List-I

List-II

Finding the Ratio $\frac{b ^{2}}{a ^{2}}$

Expressing $b$ in terms of $a$

Solving for $a$

Calculating $a^{2}$

Calculating $b^{2}$

Final Result: $a^{2} + b^{2}$