JEE Main · 2021 · 26 Feb 2021 · Shift 2 · Ellipse

Let

L

be a common tangent line to the curves

4 x^{2} + 9 y^{2} = 36

and

(2 x)^{2} + (2 y)^{2} = 31

. Then the square of the slope of the line

L

is ______

Answer: 3

Solution

The first curve is the ellipse

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

and the second is the circle

x^{2} + y^{2} = \frac{31}{4}

.
A line

y = m x + c

touches the ellipse when

c^{2} = 9 m^{2} + 4

, and it touches the circle when its distance from the centre equals the radius:

\frac{∣ c ∣}{1 + m ^{2}} = \frac{31}{4} \Rightarrow c^{2} = \frac{31}{4} (1 + m^{2})

A common tangent meets both conditions, so

9 m^{2} + 4 = \frac{31}{4} (1 + m^{2}) \Rightarrow 36 m^{2} + 16 = 31 + 31 m^{2} \Rightarrow 5 m^{2} = 15

The square of the slope of

L

3

JEE Main · 2021 · 26 Feb 2021 · Shift 2 · Integer answer