Question

express by an algebraic equation the statement that the point P(x,y) is at a distance 3 from (-7,-3)

Answer 1

Answer on Question #41426 – Math – Analytic Geometry

Question. Express by an algebraic equation the statement that the point $P(x,y)$ is at a distance 3 from $(-7,-3)$

Solution. A distance between two points $A_{0}(x_{0},y_{0})$ and $B(x_{1},y_{1})$ can be computed by the formula:

$d(A,B)=\sqrt{(x_{0}-x_{1})^{2}+(y_{0}-y_{1})^{2}}.$

If $A=P(x,y)$ and $B=(-7,-3)$, then

$d(P,B)=\sqrt{(x+7)^{2}+(y+3)^{2}}.$

Therefore, the statement that

the point $P(x,y)$ is at a distance $3$ from $(-7,-3)$

can be expressed by the following algebraic equation:

$\sqrt{(x+7)^{2}+(y+3)^{2}}=3.$

Since the expression under the square root is always non-negative, the latter equation is equivalent to the following one:

$(x+7)^{2}+(y+3)^{2}=3^{2},$

that is

$(x+7)^{2}+(y+3)^{2}=9.$

Answer. $(x+7)^{2}+(y+3)^{2}=9.$