JEE Main 2021 (22 July Shift 1)

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Animated Solution for Mathematics - Conic Sections: Let a line $L : 2 x + y = k, k > 0$ be a tangent to the hyperbola $x^{2} - y^{2} = 3$ . If $L$ is also a tangent to the parabola $y^{2} = α x$ , then $α$ is equal to:

Q: Why is the parabola tangency condition $c = A/m$? I thought it was $c = a/m$.

It is all about notation! In the standard form $y^2 = 4Ax$, the parameter is $A$. If you use the form $y^2 = 4ax$, the condition is $c = a/m$. Since our parabola is $y^2 = \alpha x$, comparing it to $y^2 = 4Ax$ means $4A = \alpha$, so $A = \alpha/4$. As long as you are consistent with your variables, the logic remains rock solid.

Select Answer:

Progressive Hints

Hint 1

Hint 1 (The Spark): Imagine the line

L

as a bridge connecting two different worlds. It is not just a line; it is a geometric constraint. Visualize the hyperbola and the parabola. The line must touch both. The key is to realize that the line's equation

2 x + y = k

is fixed by the hyperbola's tangency condition, and then that same line must satisfy the parabola's tangency condition.

Hint 2

Hint 2 (The Arsenal): You need the standard tangency conditions for both curves. For a hyperbola

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

, the condition for a line

y = m x + c

to be tangent is

c^{2} = a^{2} m^{2} - b^{2}

. For a parabola

y^{2} = 4 A x

, the condition is

c = \frac{A}{m}

. Ensure you convert your given equations into these standard forms first.

Hint 3

Hint 3 (The Execution): Start by rewriting the hyperbola

x^{2} - y^{2} = 3

\frac{x ^{2}}{3} - \frac{y ^{2}}{3} = 1

and the line as

y = - 2 x + k

. Apply the hyperbola tangency condition

c^{2} = a^{2} m^{2} - b^{2}

to find

k

. Once you have

k

, use the parabola's tangency condition

c = \frac{A}{m}

with

A = \frac{α}{4}

to solve for

α

Visualized Solution

Visualizing the Setup

Hyperbola: $x^{2} - y^{2} = 3$
Line $L : 2 x + y = k, k > 0$
Objective: Find $α$ if $L$ is tangent to both curves.

Standard Form of Hyperbola

Divide $x^{2} - y^{2} = 3$ by $3$ :
$\frac{x ^{2}}{3} - \frac{y ^{2}}{3} = 1$
Comparing with $\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$ :
$a^{2} = 3$ and $b^{2} = 3$

Slope-Intercept Form of Line

Line $L : 2 x + y = k$
Rearranging to $y = m x + c$ form:
$y = - 2 x + k$
Slope $m = - 2$ and intercept $c = k$

Tangency Condition for Hyperbola

Condition for tangency to hyperbola:
$c^{2} = a^{2} m^{2} - b^{2}$

Substituting Values

Substitute $c = k, a^{2} = 3, b^{2} = 3, m = - 2$ :
$k^{2} = 3 (- 2)^{2} - 3$

Solving for $k$

$k^{2} = 3 (4) - 3$
$k^{2} = 12 - 3 = 9$
$k = \pm 3$

Applying Constraints

Given $k > 0$ , therefore $k = 3$ .
Line $L : y = - 2 x + 3$

Introducing the Parabola

Parabola: $y^{2} = α x$
Comparing with $y^{2} = 4 A x$ :
$4 A = α \Rightarrow A = \frac{α}{4}$

Tangency Condition for Parabola

Condition for tangency to parabola:
$c = \frac{A}{m}$

Substituting Values for Parabola

Substitute $c = 3, m = - 2, A = \frac{α}{4}$ :
$3 = \frac{α /4}{- 2}$

Solving for $α$

$3 = \frac{α}{- 8}$

Final Answer

$α = 3 \times (- 8)$
$α = - 24$

00:00 / 00:00

The Sigma Insight: Standard Equations of Parabola, Ellipse, and Hyperbola

Solution Diagram

Analyzing the Hyperbola

We begin with the hyperbola:

x^{2} - y^{2} = 3

. To understand its properties, we must bring it into its standard form.

By dividing the entire equation by

3

, we transform it into:

\frac{x ^{2}}{3} - \frac{y ^{2}}{3} = 1

Here, we identify the parameters

a^{2} = 3

and

b^{2} = 3

. This standardization is the foundation of our journey, revealing the curve's orientation and scale.

The Bridge

Defining the Line

Now, consider the line

L : 2 x + y = k

. This line acts as the common thread connecting our two conics.

To analyze its interaction with the curves, we rewrite it in slope-intercept form:

y = - 2 x + k

From this, we identify the slope

m = - 2

and the

y

-intercept

c = k

. This line is destined to touch both curves, provided it satisfies their respective tangency conditions.

The Tangency Condition for the Hyperbola

For a hyperbola

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

, the condition for a line

y = m x + c

to be tangent is:

c^{2} = a^{2} m^{2} - b^{2}

Substituting our known values—

c = k

a^{2} = 3

b^{2} = 3

, and

m = - 2

—we obtain:

k^{2} = 3 (- 2)^{2} - 3

Calculating this, we find

k^{2} = 3 (4) - 3 = 12 - 3 = 9

, which implies

k = \pm 3

. Given the constraint

k > 0

, we discard the negative root and conclude that

k = 3

The Parabola's Mystery

Next, we examine the parabola:

y^{2} = α x

. Comparing this to the standard form

y^{2} = 4 A x

, we identify

4 A = α

, which gives

A = \frac{α}{4}

The condition for the line

y = m x + c

to be tangent to the parabola

y^{2} = 4 A x

is:

c = \frac{A}{m}

We know

c = 3

m = - 2

, and

A = \frac{α}{4}

. Substituting these values into the condition, we get:

3 = \frac{α /4}{- 2}

Final Calculation

With the equation

3 = \frac{α}{- 8}

, the path to the solution is clear.

Performing a simple cross-multiplication yields:

α = 3 \times (- 8) = - 24

The final value of the parameter is $α = - 24$ . We have navigated the constraints, respected the geometry, and arrived at the truth.

Common Traps & FAQs

Wait, why do we use

c^{2} = a^{2} m^{2} - b^{2}

for the hyperbola? Is it always this?

That is a fantastic question! This is the standard condition for a line

y = m x + c

to be tangent to the hyperbola

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

. If you ever forget it, you can derive it by substituting

y = m x + c

into the hyperbola equation, which gives you a quadratic in

x

. Since the line is tangent, the discriminant of this quadratic must be zero. Setting

D = 0

leads directly to this beautiful formula.

Why did we reject the negative value for

k

? The math gave me

k = \pm 3

You are absolutely right that the algebra gives two solutions,

k = 3

and

k = - 3

. However, always look back at the problem statement! The question explicitly states

k > 0

. In competitive exams like JEE, these constraints are not just filler; they are crucial filters to ensure you are paying attention to the domain of the problem. Always respect the constraints!

Why is the parabola tangency condition

c = A / m

? I thought it was

c = a / m

It is all about notation! In the standard form

y^{2} = 4 A x

, the parameter is

A

. If you use the form

y^{2} = 4 a x

, the condition is

c = a / m

. Since our parabola is

y^{2} = α x

, comparing it to

y^{2} = 4 A x

means

4 A = α

, so

A = α /4

. As long as you are consistent with your variables, the logic remains rock solid.

Animated Solution for Mathematics - Conic Sections: Let a line $L : 2 x + y = k, k > 0$ be a tangent to the hyperbola $x^{2} - y^{2} = 3$ . If $L$ is also a tangent to the parabola $y^{2} = α x$ , then $α$ is equal to:

Select Answer:

Visualized Solution

Visualizing the Setup

Standard Form of Hyperbola

Slope-Intercept Form of Line

Tangency Condition for Hyperbola

Substituting Values

Solving for $k$

Applying Constraints

Introducing the Parabola

Tangency Condition for Parabola

Substituting Values for Parabola

Solving for $α$

Final Answer

The Sigma Insight: Standard Equations of Parabola, Ellipse, and Hyperbola

Analyzing the Hyperbola

The Bridge

The Tangency Condition for the Hyperbola

The Parabola's Mystery

Final Calculation

Similar Questions

If the line $α x + 2 y = 1$ , where $α \in R$ , does not meet the hyperbola $x^{2} - 9 y^{2} = 9$ , then a possible value of $α$ is:

A line is a common tangent to the circle $(x - 3)^{2} + y^{2} = 9$ and the parabola $y^{2} = 4 x$ . If the two points of contact $(a, b)$ and $(c, d)$ are distinct and lie in the first quadrant, then $2 (a + c)$ is equal to

Equation of a common tangent to the parabola $y^{2} = 4 x$ and the hyperbola $x y = 2$ is :

Let P be the point of intersection of the common tangents to the parabola $y^{2} = 12 x$ and the hyperbola $8 x^{2} - y^{2} = 8$ . If S and S' denote the foci of the hyperbola where S lies on the positive x-axis then P divides SS' in a ratio:

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Select Answer:

Visualized Solution

Visualizing the Setup

Standard Form of Hyperbola

Slope-Intercept Form of Line

Tangency Condition for Hyperbola

Substituting Values

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

Applying Constraints

Introducing the Parabola

Tangency Condition for Parabola

Substituting Values for Parabola

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

Final Answer

The Sigma Insight: Standard Equations of Parabola, Ellipse, and Hyperbola

Analyzing the Hyperbola

The Bridge

The Tangency Condition for the Hyperbola

The Parabola's Mystery

Final Calculation

Similar Questions

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