Question

If $F_1$ and $F_2$ are the foci of a hyperbola, and $P$ is a point on one of its branches, then the tangent to the curve at $P$, bisects the angle $F_{1}P{F}_2$ in the ratio

• A. $2:1$
• B. $1:1$
• C. $3:1$
• D. $4:1$

Answer 1

The correct option is B $1:1$

Consider the figure

Let $C$ be the origin, $C{F}_{1}X$, the axis of $X$ and a straight line $CY$ through $C$ perpendicular to $CX$ as the axis of $Y$, here $F_1$ and $F_2$ are the foci of a hyperbola, $P$ is a point on one of its branches. Let say $P(x,y)$ be any point on the hyperbola and $PM$ and $PN$ be the perpendiculars from $P$ on $kz$ and $kx$

$\therefore SP=ePM{SP}^2=e^2{PM}^2{SP}^2=e^2{kn}^2=e^2{(CN-Ck)}^2{(x-ae)}^2+y^2=e^2{[x-\frac{a}{e}]}^2\therefore x^2(e^2-1)-y^2=a^2(e^2-1)\frac{x^2}{a^2}-\frac{y^2}{a^2(e^2-1)}=1\therefore \frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

$\therefore$ the angle $F_{1}P{F}_2$ will be in ratio $1:1$