Question

The circle whose equation is (x+3)² + (y + 4)² = 17 and the line whose equation is y = 4x + 25 are tangent. Determine the point of tangency,

Answer 1

Answer:

(-7,-3)

Explanation:

The equation of the circle is: (x+3)² + (y + 4)² = 17

The equation of the line is: y=4x+25

The point of tangency is the point where the circle and the line meets.

$\mleft(x+3\mright)^2+(y+4)^2=17$

Substituting the linear equation into the above, we have:

$\begin{gathered} (x+3)^2+(4x+25+4)^2=17 \\ (x+3)^2+(4x+29)^2=17 \end{gathered}$

We solve for x.

$\begin{gathered} x^2+6x+9+16x^2+232x+841=17 \\ x^2+16x^2+6x+232x+9+841-17=0 \\ 17x^2+238x+833=0 \\ 17(x^2+14x+49)=0 \\ x^2+14x+49=0 \\ x^2+7x+7x+49=0 \\ x(x+7)+7(x+7)=0 \\ (x+7)(x+7)=0 \\ x=-7(\text{twice)} \end{gathered}$

Next, we solve for y.

$\begin{gathered} y=4x+25 \\ y=4(-7)+25 \\ =-28+25 \\ y=-3 \end{gathered}$

The point of tangency is (-7,-3).