Find the condition under which the line $xcosalpha+ysinalpha=p$ will be a tangent to the conic $3x^2+4y^2=5$ ?

Question

Find the condition under which the line $xcosalpha+ysinalpha=p$
will be a tangent to the conic
$3x^2+4y^2=5$

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Answer 1 · 2017-03-15T20:19:50

See below.

Given $f(x,y)=3x^2+4y^2-5$ the normal to the curve $f(x.y)=0$ at point $p_0=(x_0,y_0)$ is

$vec n_0=(f_x,f_y)(x_0,y_0) = (6x_0,8y_0) = 2(3x_0,4y_0)$

Calling $p = (x,y)$ a tangent to $f(x,y)=0$ at point $x_0,y_0$ is given by

$l-> << p-p_0, vec n_0 >> =0$

where $<< cdot, cdot >>$ denotes the scalar product of two vectors, or

$l->6 x x_0 + 8 y y_0 - 2(3 x_0^2 + 4 y_0^2)=0$ or

$l->6 x x_0 + 8 y y_0 - 2 cdot 5=0$ or

$l->3 x x_0 +4 y y_0 - 5=0$ and also

$l-> x((3x_0)/sqrt((3x_0)^2+(4y_0)^2))+y((4y_0)/sqrt((3x_0)^2+(4y_0)^2))=5/sqrt((3x_0)^2+(4y_0)^2)$

Now, calling

$cosalpha = (3x_0)/sqrt((3x_0)^2+(4y_0)^2)$
$sinalpha = (4y_0)/sqrt((3x_0)^2+(4y_0)^2)$
$p = 5/sqrt((3x_0)^2+(4y_0)^2)$

or

$cosalpha = (3 x_0)/sqrt[20 - 3 x_0^2]$
$sin alpha = 2sqrt((3x_0^2-5)/(3x_0^2-20)$
$p=5/sqrt[20 - 3 x_0^2]$

we have the sough tangent line