Question

Compute the shortest distance between the circle $x^2+y^2-10x-14y-151=0$ and the point $(-7,2)$.

• A. $0$
• B. $1$
• C. $2$
• D. $4$

Answer 1

The correct option is C

$2$

Explanation for the correct option:

Step 1: Find the centre and radius of the circle:

Compare the given circle with the standard equation of the circle to get the centre and radius.

Given: $x^2+y^2-10x-14y-151=0$

Centre will be negative of coefficient of $\left(x,y\right)$

$\begin{array}{rcl}C& =& \left(-\left(-5\right),-\left(-7\right)\right)\\ & =& \left(5,7\right)\end{array}$

Also, the radius is estimated using the coefficient of $\left(x,y\right)$ and the constant.

$\begin{array}{rcl}r& =& \sqrt{\left(5^2+7^2\right)+151}\\ & =& \sqrt{25+49+151}\\ & =& \sqrt{225}\\ & =& 15\end{array}$ $\left[\because r=\sqrt{g^2+f^2-c}\right]$

Step 2: Find the distance between the centre and the given point:

Use the distance formula to estimate the distance between the points.

$\begin{array}{rcl}d& =& \sqrt{{\left(5-\left(-7\right)\right)}^2+{\left(7-2\right)}^2}\\ & =& \sqrt{144+25}\\ & =& \sqrt{169}\\ & =& 13\end{array}$ $\left[\because d=\sqrt{{\left(y_2-y_1\right)}^2+{\left(x_2+x_1\right)}^2}\right]$

Step 3: Find the shortest distance between the point and the circle:

Use the relation that the shortest distance will be the distance between centre and radius deducted from radius.

$\begin{array}{rcl}s& =& 15-13\\ & =& 2\end{array}$

Hence, option (C) is the correct answer.