Question

Assertion :The length of subnormal to $S=\frac{x^2}{a^2}-\frac{y^2}{b^2}-1$ at the point $(x_1,y_1)$ is $\frac{b^2{x}_1}{a^2}$ Reason: The length of subtangent to $S=\frac{x^2}{a^2}-\frac{y^2}{b^2}-1$ at the point $(x_1,y_1)$ is $x_1-\frac{a^2}{x_1}$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

For any standard Hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

Let the Normal at point $P(x_1,y_1)$ meet the $x$-axis at G$(x,0)$,

Let the tangent at point $P(x_1,y_1)$ meet the $x$-axis at $T(x,0)$,

$C(0,0)$ is the center and the perpendicular drawn from point $P$ to $x-$axis meets the $x-$ axis at $N$

The coordinates of point $N$ are $(x_1,0)$, hence $CN=x_1$

The equation of normal at $P(x_1,y_1)$ is $\frac{a^{2}x}{x_1}+\frac{b^{2}y}{y_1}=a^2{e}^2$.....$\left(1\right)$

Point $G$ lies on $x-$axis so its $y$ ordinate is $0$

For finding $x-$ ordinate, putting value of $y=0$ in eq. $\left(1\right)$ we get,

$\to$ $x=e^2{x}_1$

So point $G$ is $(e^2{x}_1,0)$ and $CG=e^2{x}_1$

From the figure, we can know that the length of sub-normal for a standard hyperbola is $NG$

Also from figure $NG=CG-CN$

As $C$ is $(0,0)$, $N$ is $(x_1,0)$ and $G$ is $(e^2{x}_1,0)$

$\because$ All the points lie on $x$-axis, in a line.

$\to$ $NG=(e^2-1)x_1$, or $NG=(1+\frac{b^2}{a^2}-1)x_1$, $\because e^2=1+\frac{b^2}{a^2}$

$\to NG=\frac{b^2}{a^2}{x}_1$

Hence the length of subnormal is $NG=\frac{b^2}{a^2}{x}_1$,

$Soassertioniscorrect.$

Also, the equation of tangent at $P(x_1,y_1)$ is $\frac{x{x}_1}{a^2}-\frac{y{y}_1}{b^2}=1$.....$\left(1\right)$

Point $T$ lies on $x-$axis so its $y$ ordinate is $0$

For finding $x-$ ordinate, putting value of $y=0$ in eq. $\left(1\right)$ we get,

$\to$ $x=\frac{a^2}{x_1}$

So point $T$ is $(\frac{a^2}{x_1},0)$ and $CT=\frac{a^2}{x_1}$

From the figure, we can know that the length of sub-tangent for a standard hyperbola is $TN$

Also from figure $TN=CN-CT$

$\because$ All the points lie on $x$-axis, in a line.

$\to$ $TN=x_1-\frac{a^2}{x_1}$,

Hence the length of subtangent is $TN=x_1-\frac{a^2}{x_1}$

$Soreasonisalsocorrect.$

So we can see the assertion and the reason both are correct, but the length of subnormal of a hyperbola doesn't correlate with length of subtangent of the hyperbola, so the reason is not a correct explanation of assertion. Hence option $B$ is correct.