NDA 2024 Maths · Mathematics · Conic Sections · hard

The line y=x+k touches the ellipse (x^2/(9)+(y^2/(4)=1 for k equal to

Single correct option

The line

y=x+k

touches the ellipse

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

for

k

equal to

Show answer and solution

Answer: (A) $\pm\sqrt{13}$

For the ellipse

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

the line

y=mx+c

is tangent iff

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

. Here

m=1

, so

k^2=9+4=13

and

k=\pm\sqrt{13}

. Trap: subtracting (

9-4=5

, which is

c^2

of the foci) or using the circle condition gives the other options.

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