MaplePrimes - answers and comments on Question, Tangent line to a circle through an external point

Standard calculus problem

rlopez — Tue, 20 Nov 2012 22:37:53 Z

The point of contact of the tangent line must be a point on the circle.

The slope of the tangent line is the slope of the circle at the point of contact.

Two conditions, and two equations:

x^2+y^2=4

y=-x/y*(x-4)+1

The second equation is the equation of a tangent line. The slope is obtained by implicit differentiation applied to the circle. The x and y in the expression for the slope will be the coordinates of the point of contact. The x and y used for the equation of the line must also be the coordinates of the point of contact.

Solve these two equations simultaneously, and the two points of contact will be returned. From this, it should be simple to write the equations for the two tangent lines.

RJL Maplesoft

Solution scheme

Kitonum — Tue, 20 Nov 2012 22:47:22 Z

1) Find the equation of the tangent line at any point (x0, y0) of the circle.

2) Write down the condition that the tangent line passes through the point (4,1).

3) Solving the resulting system of two equations with two unknowns x0 and y0, find the point of contact, and then the tangent line. Get two solutions.

Another way

toandhsp — Wed, 21 Nov 2012 11:19:53 Z

Put the equation of the line passing the point A(4,1) has the form a*(x-4)+b*(y-1)=0. We always assume a^2 + b^2 = 1. And then, you solve the system of equations

solve([-4*a-b=2,a^2+b^2=1],[a,b]);

But I don't know how to get the exactly solutions [a = -8/17-1/17*13^(1/2),b = -2/17+4/17*13^(1/2)] and [[a = -8/17+1/17*13^(1/2), b = -2/17-4/17*13^(1/2)]]

I must solve in two cases

solve([f(0,0)=2,a^2+b^2=1,b<0],[a,b]);

solve([f(0,0)=2,a^2+b^2=1,b>0],[a,b]);

Note that, when the line passing the point A(2,1), the line x = 2 has not slope, but it is also a tangent of the circle. You try

f:=(x,y)->a*(x-2)+b*(y-1):

solve([f(0,0)=2,a^2+b^2=1],[a,b]);

helpFile

herclau — Wed, 21 Nov 2012 22:50:12 Z

Tangentlinrev.mw

why the command does not work: _EnvHorizontalName := x,_EnvVerticalName := y ?
And a few lines later asks me the name of the axes

restart;
with(plots); with(plottools);

The circle S with center at the origin and radius 2 is given by the equation x2 + y2 = 4
The circle S have two tangents passing through the point (4, 1). Find the equation for each of them.
x^2+y^2 = 4;

subs(y(x) = y, solve(diff(subs(y = y(x), x^2+y^2 = 4), x), diff(y(x), x)));
y-1 = -x*(x-4)/y;
x^2+y^2 = 4, y-1 = -x*(x-4)/y;

solve({y-1 = -x*(x-4)/y, x^2+y^2 = 4}, {x, y});

allvalues({x = 1-RootOf(17*_Z^2-2*_Z-3), y = 4*RootOf(17*_Z^2-2*_Z-3)});

cp1 := eval([x, y], %[1]);
cp2 := eval([x, y], `%%`[2]);

graph1 := implicitplot(x^2+y^2 = 4, x = -3 .. 3, y = -3 .. 3);
extp := point([4, 1], color = black, symbol = cross, symbolsize = 25);
ccp1 := point(cp1, color = green, symbolsize = 25, symbol = circle);
ccp2 := point(cp2, color = green, symbolsize = 25, symbol = circle);
tl1 := line([4, 1], cp1, color = red, linestyle = solid, color = blue);
tl2 := line([4, 1], cp2, color = red, linestyle = solid, color = blue);

display([graph1, extp, ccp1, ccp2, tl1, tl2]);

with(geometry);
_EnvHorizontalName := x;_EnvVerticalName := y;

point(Ex, 4, 1), point(A, cp1[1], cp1[2]), point(B, cp2[1], cp2[2]);

line(t1, [Ex, A]);
line(t2, [Ex, B]);
map(detail, [t1, t2]);

Equation(t1);
Equation(t2);

ll1 := implicitplot(Equation(t1), x = -3 .. 5, y = -3 .. 3, color = black);
ll2 := implicitplot(Equation(t2), x = -3 .. 5, y = -3 .. 3, color = black);
display([graph1, extp, ccp1, ccp2, ll1, ll2]);

Gracias