Name: Orthogonal Curves
Uploaded: 2019-07-19
Duration: 11 min 48 s
Description: Have you heard the term 'orthogonal curves'? Can you find angles between two curves? Do watch this video and find all such answers!! Have fun!!

Question

From an external point

P

, tangents are drawn to the parabola,

y^{2} = 4 a x

; find the equation of the locus of

P

when these tangents make angles

θ_{1}

and

θ_{2}

with the axis, such that

\cos θ_{1} \cos θ_{2}

is a constant

(= μ) .

Accepted Answer

The equation of the parabola is $y^{2} = 4 a x . . . (1)$ .

The equation of the tangent with given slope $(m)$ is given as $y = m x + \frac{a}{m} . . . (2)$ .

If $P (h, k)$ is the external point from which the tangents are drawn, then the coordinates of $P$ will satisfy the equation $(2)$ .

This means that we have: $k = m h + \frac{a}{m}$ or, $m^{2} h - m k + a = 0 . . . (3)$ .

Now, since equation $(3)$ is quadratic in $m$ , so we have

$m_{1} + m_{2} = \frac{k}{h} . . . (4)$

and, $m_{1} m_{2} = \frac{a}{h} . . . (5)$ .

Given, $θ_{2}$
Or, $\sec^{2} θ_{1} \sec^{2} θ_{2} = \frac{1}{μ^{2}}$
Or, $\sec^{2} θ_{1} \sec^{2} θ_{2} = \frac{1}{μ^{2}}$
Or, $(1 + \tan^{2} θ_{1}) (1 + \tan^{2} θ_{2}) = \frac{1}{μ^{2}}$
Or, $1 + \tan^{2} θ_{1} + \tan^{2} θ_{2} + \tan^{2} θ_{1} \times \tan^{2} θ_{2} = \frac{1}{μ^{2}}$
Or, $1 + {(\tan θ_{1} + \tan θ_{2})}^{2} - 2 \tan θ_{1} \tan θ_{2} + \tan^{2} θ_{1} \tan^{2} θ_{2} = \frac{1}{μ^{2}}$
From equations $(4)$ and $(5)$ , we get
$1 + \frac{k^{2}}{h^{2}} - \frac{2 a}{h} + {(\frac{a}{h})}^{2} = \frac{1}{μ^{2}}$
Or, $h^{2} = μ^{2} \{{(h - a)}^{2} + k^{2}\}$
Generalizing the locus of $P (h, k)$
$x^{2} = μ^{2} \{{(x - a)}^{2} + y^{2}\}$

From an external point $P$ , tangents are drawn to the parabola, $y^{2} = 4 a x$ ; find the equation of the locus of $P$ when these tangents make angles $θ_{1}$ and $θ_{2}$ with the axis, such that $\cos θ_{1} \cos θ_{2}$ is a constant $(= μ) .$

Important Questions on Parabola

Learn and Prepare for any exam you want !

From an external point P, tangents are drawn to the parabola, y2=4ax; find the equation of the locus of P when these tangents make angles θ1 and θ2 with the axis, such that cos⁡θ1cos⁡θ2 is a constant (=μ).

Important Questions on Parabola

Learn and Prepare for any exam you want !

From an external point $P$ , tangents are drawn to the parabola, $y^{2} = 4 a x$ ; find the equation of the locus of $P$ when these tangents make angles $θ_{1}$ and $θ_{2}$ with the axis, such that $\cos θ_{1} \cos θ_{2}$ is a constant $(= μ) .$