JEE Main 2020 (7 Jan Evening)

LEVELJEE Main

Animated Solution for Mathematics - Conic Sections: If $3 x + 4 y = 122$ is a tangent to the ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{9} = 1$ for some $a \in R$ , then the distance between the foci of the ellipse is:

Select Answer:

Visualized Solution

Analyze the Given Tangent Line

Given tangent line: $3 x + 4 y = 122$
We need to convert this to the standard slope-intercept form: $y = m x + c$

Find Slope $m$ and Intercept $c$

Rearranging: $4 y = - 3 x + 122$
Dividing by $4$ : $y = - \frac{3}{4} x + 32$
Comparing with $y = m x + c$ :
$m = - \frac{3}{4}$ and $c = 32$

Identify Ellipse Parameters

Ellipse equation: $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{9} = 1$
Standard form: $\frac{x ^{2}}{A ^{2}} + \frac{y ^{2}}{B ^{2}} = 1$
By comparison: $A^{2} = a^{2}$ and $B^{2} = 9$

The Condition of Tangency

For a line $y = m x + c$ to be tangent to $\frac{x ^{2}}{A ^{2}} + \frac{y ^{2}}{B ^{2}} = 1$ :
The condition is: $c^{2} = A^{2} m^{2} + B^{2}$

Substitute Values into the Condition

We know: $m = - \frac{3}{4}$ , $c = 32$ , $A^{2} = a^{2}$ , $B^{2} = 9$
Substituting into $c^{2} = A^{2} m^{2} + B^{2}$ :
$(32)^{2} = a^{2} (- \frac{3}{4})^{2} + 9$

Simplify and Solve for $a^{2}$

Squaring the terms: $18 = a^{2} (\frac{9}{16}) + 9$
Subtracting $9$ from both sides: $9 = a^{2} (\frac{9}{16})$
Solving for $a^{2}$ : $a^{2} = 16$
Therefore, $a = 4$

Introduction to Eccentricity

Since $a^{2} = 16$ and $b^{2} = 9$ , we have $a > b$ .
The formula for eccentricity $e$ is:
$e = 1 - \frac{b ^{2}}{a ^{2}}$

Calculate Eccentricity $e$

Substituting $a^{2} = 16$ and $b^{2} = 9$ :
$e = 1 - \frac{9}{16}$
$e = \frac{16 - 9}{16}$
$e = \frac{7}{16} = \frac{7}{4}$

Distance Between Foci Formula

The foci of the ellipse are located at $(\pm a e, 0)$ .
The distance between the two foci is given by:
Distance $= 2 a e$

Final Calculation

We have $a = 4$ and $e = \frac{7}{4}$ .
Substituting into the distance formula:
Distance $= 2 \times 4 \times \frac{7}{4}$
Distance $= 27$

Conclusion and Key Takeaway

Condition of tangency: $c^{2} = a^{2} m^{2} + b^{2}$ is essential for finding unknown ellipse parameters.
Eccentricity: $e = 1 - \frac{b ^{2}}{a ^{2}}$ connects the axes lengths.
Focal Distance: The distance between foci is always $2 a e$ .
Final Answer: $27$

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The Sigma Insight: Equation of Tangent and Normal

Solution Diagram

Analyzing the Setup

We are given the equation of a line

3 x + 4 y = 122

and the equation of an ellipse

\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{9} = 1

. Our objective is to determine the distance between the foci of this ellipse.

To begin, we transform the line equation into the slope-intercept form,

y = m x + c

. Starting with

4 y = - 3 x + 122

, we divide by

4

to obtain:

y = - \frac{3}{4} x + 32

From this, we identify the slope

m = - \frac{3}{4}

and the

y

-intercept

c = 32

The Bridge of Tangency

For a line

y = m x + c

to be tangent to an ellipse

\frac{x ^{2}}{A ^{2}} + \frac{y ^{2}}{B ^{2}} = 1

, it must satisfy the condition of tangency:

c^{2} = A^{2} m^{2} + B^{2}

Substituting our known values

A^{2} = a^{2}

B^{2} = 9

m = - \frac{3}{4}

, and

c = 32

into this condition, we get:

(32)^{2} = a^{2} (- \frac{3}{4})^{2} + 9

Simplifying the squares, we have

18 = a^{2} (\frac{9}{16}) + 9

. Subtracting

9

from both sides yields

9 = a^{2} (\frac{9}{16})

, which simplifies to $a^{2} = 16$ , or $a = 4$ .

The Heart of the Ellipse

With

a = 4

and

b = 3

(since

b^{2} = 9

), we observe that

a > b

, confirming the ellipse is elongated along the

x

-axis. We now calculate the eccentricity

e

using the formula:

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

Substituting the values, we find:

e = 1 - \frac{9}{16} = \frac{7}{16} = \frac{7}{4}

The foci of the ellipse are located at

(\pm a e, 0)

. Therefore, the distance between the two foci is given by

2 a e

Calculating this distance:

2 a e = 2 \times 4 \times \frac{7}{4} = 27

The final distance between the foci is $27$ .

Common Traps & FAQs

Why do we use the condition

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

? Can't we just use the discriminant method?

You absolutely can! Setting the discriminant of the quadratic equation (formed by substituting the line into the ellipse) to zero is the fundamental derivation. However, in competitive exams like JEE, memorizing the standard condition

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

saves precious time and reduces calculation errors. It's a tool in your belt, not a replacement for understanding.

How do I know if

a > b

b > a

Great question. Look at the denominators. If the denominator under

x^{2}

is larger, the major axis is along the x-axis (

a > b

). If the denominator under

y^{2}

is larger, the major axis is along the y-axis (

b > a

). In this problem, we found

a^{2} = 16

and

b^{2} = 9

, so

16 > 9

, confirming the major axis is horizontal.

Is the distance between foci always

2 a e

Yes, for any standard ellipse centered at the origin, the foci are at

(\pm a e, 0)

if the major axis is horizontal, or

(0, \pm b e)

if vertical. The distance between them is the magnitude of the difference of their coordinates, which is

2 a e

(or

2 b e

depending on orientation).

Animated Solution for Mathematics - Conic Sections: If $3 x + 4 y = 122$ is a tangent to the ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{9} = 1$ for some $a \in R$ , then the distance between the foci of the ellipse is:

Select Answer:

Visualized Solution

Analyze the Given Tangent Line

Find Slope $m$ and Intercept $c$

Identify Ellipse Parameters

The Condition of Tangency

Substitute Values into the Condition

Simplify and Solve for $a^{2}$

Introduction to Eccentricity

Calculate Eccentricity $e$

Distance Between Foci Formula

Final Calculation

Conclusion and Key Takeaway

The Sigma Insight: Equation of Tangent and Normal

Analyzing the Setup

The Bridge of Tangency

The Heart of the Ellipse

Similar Questions

If $3 x + 4 y = 122$ is a tangent to the ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{9} = 1$ , for some $a \in R$ then the distance between the foci of the ellipse is :

Let the tangents at the points $P$ and $Q$ on the ellipse $\frac{x ^{2}}{2} + \frac{y ^{2}}{4} = 1$ meet at the point $R (2, 22 - 2)$ . If $S$ is the focus of the ellipse on its negative major axis, then $S P^{2} + S Q^{2}$ is equal to

If the line $α x + 4 y = 7$ , where $α \in R$ , touches the ellipse $3 x^{2} + 4 y^{2} = 1$ at the point $P$ in the first quadrant, then one of the focal distances of $P$ is :

On the ellipse $4 x^{2} + 9 y^{2} = 1$ , the points at which the tangents are parallel to the line $8 x = 9 y$ are

Let $E$ be an ellipse whose axes are parallel to the co-ordinates axes, having its center at $(3, - 4)$ , one focus at $(4, - 4)$ and one vertex at $(5, - 4)$ . If $m x - y = 4, m > 0$ is a tangent to the ellipse $E$ , then the value of $5 m^{2}$ is equal to

If the tangents on the ellipse $4 x^{2} + y^{2} = 8$ at the points $(1, 2)$ and $(a, b)$ are perpendicular to each other, then $a^{2}$ is equal to :

If the normal to the ellipse $3 x^{2} + 4 y^{2} = 12$ at a point P on it is parallel to the line, $2 x + y = 4$ and the tangent to the ellipse at P passes through Q(4, 4) then PQ is equal to :

Select Answer:

Visualized Solution

Analyze the Given Tangent Line

Find Slope .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } m and Intercept .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } c

Identify Ellipse Parameters

The Condition of Tangency

Substitute Values into the Condition

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

Introduction to Eccentricity

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

Distance Between Foci Formula

Final Calculation

Conclusion and Key Takeaway

The Sigma Insight: Equation of Tangent and Normal

Analyzing the Setup

The Bridge of Tangency

The Heart of the Ellipse

Similar Questions

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If 3x+4y=122​ is a tangent to the ellipse a2x2​+9y2​=1, for some a∈R then the distance between the foci of the ellipse is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the line αx+4y=7​, where α∈R, touches the ellipse 3x2+4y2=1 at the point P in the first quadrant, then one of the focal distances of P is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } On the ellipse 4x2+9y2=1, the points at which the tangents are parallel to the line 8x=9y are

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the tangents on the ellipse 4x2+y2=8 at the points (1,2) and (a,b) are perpendicular to each other, then a2 is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the normal to the ellipse 3x2+4y2=12 at a point P on it is parallel to the line, 2x+y=4 and the tangent to the ellipse at P passes through Q(4, 4) then PQ is equal to :

Find Slope $m$ and Intercept $c$

Simplify and Solve for $a^{2}$

Calculate Eccentricity $e$

If $3 x + 4 y = 122$ is a tangent to the ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{9} = 1$ , for some $a \in R$ then the distance between the foci of the ellipse is :

If the line $α x + 4 y = 7$ , where $α \in R$ , touches the ellipse $3 x^{2} + 4 y^{2} = 1$ at the point $P$ in the first quadrant, then one of the focal distances of $P$ is :

On the ellipse $4 x^{2} + 9 y^{2} = 1$ , the points at which the tangents are parallel to the line $8 x = 9 y$ are

If the tangents on the ellipse $4 x^{2} + y^{2} = 8$ at the points $(1, 2)$ and $(a, b)$ are perpendicular to each other, then $a^{2}$ is equal to :

If the normal to the ellipse $3 x^{2} + 4 y^{2} = 12$ at a point P on it is parallel to the line, $2 x + y = 4$ and the tangent to the ellipse at P passes through Q(4, 4) then PQ is equal to :