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Hyperbola

Section: Mathematics

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Question : 46 of 98

Marks: +1, -0

A common tangent

T

C_1: \;\frac{x^2}{4}+\;\frac{y^2}{9}=1

C_2: \;\frac{x^2}{42}-\;\frac{y^2}{143}=1

does not pass through the fourth quadrant. If

T

C_1

(x_1, y_1)

C_2

(x_2, y_2)

2 x_1+x_2|

is equal to ________.

[27-Jul-2022-Shift-2]

Your Answer:

Solution:

Equation of tangent to ellipse

\;\frac{x^2}{4}+\;\frac{y^2}{9}=1

and given slope

m

is :

y=m x+\sqrt{4 m^2+9} \ldots

. (i)

For slope m equation of tangent to hyperbola is :

y=m x+\sqrt{42 m^2-143}

Tangents from (i) and (ii) are identical then

4 m^2+9=42 m^2-143

\therefore m=\pm 2(+2 \;\text{ is not acceptable }\;)

\therefore m=-2 .

Hence,

x_1=\;\frac{8}{5}

and

x_2=\;\frac{84}{5}

\therefore |2 x_1+x_2|=\left|\;\frac{16}{5}+\;\frac{84}{5}\right|=20

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