Answer to Question #248583 in Calculus for Anne

Question

At what point(s) on the circle x
2 + y
2 = 13 is its tangent line parallel
to the line 3x ⠈ ’ 2y =

Answer 1

Solution:

Let the given circle be $x^2+y^2=13$ ...(i)

And line is $3x-2y=6$ ...(ii)

Differentiate (i) w.r.t. $x$

$2x+2y.y'=0 \\\Rightarrow x+y.y'=0 \\\Rightarrow y'=-\dfrac xy$

Slope of line in (ii) is $m=\dfrac32$

Since, given line is parallel to the tangent, so their slopes are equal.

$y'=m \\\Rightarrow -\dfrac xy=\dfrac32 \\\Rightarrow x=-\dfrac 32y\ ...(iii)$

Put (iii) in (i)

$(\dfrac 32y)^2+y^2=13 \\\Rightarrow \dfrac 94y^2+y^2=13 \\\Rightarrow \dfrac{13}4y^2=13 \\\Rightarrow y^2=4 \\\Rightarrow y=\pm2$

Put these values of y in (iii)

When y=2, x=-3

When y=-2, x=3

Thus, points are $(-3,2),(3,-2)$.