In the following figure, `AS = 4` and `SP = 9`. The value of SZ is

Options

A. 6
B. 5.5
C. 6.5
D. none of these

Answer

A

Explanation

Let `P(at^(2),2at)` be any point on the parabola `y^(2) = 4ax`, then the equation of the tangent at P is `yt = x +at^(2)`. It cuts y-axis at (0,at).

Clearly, SZ is perpendicular to PT.
Now, `SZ = sqrt(a^(2) +a^(2)t^(2)) = a sqrt(1+t^(2))`
`Sp = a + at^(2)` and `AS = a`
`:. SZ = a^(2) (1+t^(2))` and `AS. SP = a^(2) (t^(2) +1)`
Clearly, `SZ^(2) = AS. SP`
So, from the figure, we have
`rArr SZ^(2) = (4)(9) rArr SZ = 6`