ParabolaHardBloom L3
Question
Two mutually perpendicular tangents of the parabola $y^{2} = 4ax$ meet its axis in $P_{1}$ and $P_{2}$. If $S$ is the focus of the parabola then $\frac{1}{SP_{1}} + \frac{1}{SP_{2}}$ is equal to :
Options
A.$\frac{4}{a}$
B.$\frac{2}{a}$
C.$\frac{1}{a}$
D.$\frac{1}{4a}$
Solution
Sol. [Key Concept: Tangent at $t$ meets the axis with $SP = a(1 + t^{2})$; perpendicular tangents have $t_{1}t_{2} = - 1$, giving $\frac{1}{{SP}_{1}} + \frac{1}{{SP}_{2}} = \frac{1}{a}$.]
$\begin{matrix} \frac{1}{a + at_{1}^{2}} + \frac{1}{a + at_{2}^{2}} & \ = \frac{1}{a + at_{1}^{2}} + \frac{1}{a + a\left( - \frac{1}{t_{1}} \right)^{2}} \\ & \ \end{matrix}$
$$= \frac{1}{a}$$
Create a free account to view solution
View Solution FreeMore Parabola Questions
If the normal to a parabola $y^{2} = 4ax$ at $P$ meets the curve again in $Q$ and if $PQ$ and the normal at $Q$ makes an...If the point $\left( sin\theta,\frac{1}{\sqrt{2}} \right)$ lies exterior to both the parabolas $y^{2} = |x|$, then $\the...Length of the intercept on the normal at the point $P\left( at^{2},2at \right)$ of the parabola $y^{2} = 4ax$ made by th...A parabola passes through (1, 2) and (3, 4). The tangents drawn at these points intersect at (-6, 8), then the slope of ...PQ is a normal chord of the parabola y2 = 4ax at P, A being the vertex of the parabola. Through P a line is drawn parall...