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