NIMCET 2010 — Mathematics PYQ

The correct option is (b) $\sqrt{a^2+b^2}$.

1. Equation of the Tangent:

The equation of a tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ in slope form is:

2. Intercepts on Axes:

The question states that the tangent intercepts equal lengths $l$ on both the $x$ and $y$ axes.

• For a line to have equal intercepts (ignoring the sign), the slope $m$ must be $\pm 1$.

• Let's take $m=-1$ (for a line like $x+y=l$).

3. Finding the Intercept $l$:

Substitute $m=-1$ into the tangent equation:

4. Comparing with Intercept Form:

The intercept form of a line is $\frac{x}{d}+\frac{y}{d}=1$, which can be written as $x+y=d$, where $d$ is the intercept length.

Comparing $x+y=\sqrt{a^2+b^2}$ with the intercept length $l$:

Thus, the length of the intercept $l$ is $\sqrt{a^2+b^2}$.