Let L be a common tangent line to the curves and . Then, the square of the slope of the line L is

Question

Let L be a common tangent line to the curves $\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1$ and $x^{2} + y^{2} = \frac{31}{4}$ . Then, the square of the slope of the line L is ________

Smart Achievers · Accepted Answer

(3)

Tangent to the curve $\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1$ is

$y^{2} = m x \pm \sqrt{a^{2} m^{2} + b^{2}}, y = m x \pm \sqrt{9 m^{2} + 4}$

And equation of tangent to the curve $x^{2} + y^{2} = \frac{31}{4}$ is

$y = m x \pm a \sqrt{1 + m^{2}}, y = m x \pm \sqrt{\frac{31}{4} (1 + m^{2})}$

For common tangent $9 m^{2} + 4 = \frac{31}{4} + \frac{31}{4} m^{2} \Rightarrow \frac{5}{4} m^{2} = \frac{15}{4} \Rightarrow m^{2} = 3$

Solution

Q.
Let L be a common tangent line to the curves $\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1$ and $x^{2} + y^{2} = \frac{31}{4}$ . Then, the square of the slope of the line L is ________

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Solution

Q. Let L be a common tangent line to the curves x29+y24=1 and x2+y2=314. Then, the square of the slope of the line L is ________

Ans. (3) Tangent to the curve x29+y24=1 is y2=mx±a2m2+b2, y=mx±9m2+4 And equation of tangent to the curve x2+y2=314 is y=mx±a1+m2, y=mx±314(1+m2) For common tangent 9m2+4=314+314m2⇒54m2=154⇒m2=3

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Q.
Let L be a common tangent line to the curves $\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1$ and $x^{2} + y^{2} = \frac{31}{4}$ . Then, the square of the slope of the line L is ________