Question

If $p\text{and}q$ be the longest distance and the shortest distance respectively of the point $\left(–7,2\right)$ from any point $\left(\alpha ,\beta \right)$ on the curve whose equation is $x^2+y^2–10x–14y–51=0$, then G.M. of $p\text{and}q$ is equal to

• A. $2\sqrt{11}$
• B. $5\sqrt{5}$
• C. $13$
• D. None of these

Answer 1

The correct option is A

$2\sqrt{11}$

Explanation for the correct option

Step 1: Information required for the solution

We know that the equation of the given curve is $x^2+y^2–10x–14y–51=0$ which is same as the general form of the equation of a circle that is $x^2+y^2+2hx+2ky+c=0$. After the comparison, we have

$\begin{array}{rcl}h& =& -5\end{array}$ and $k=-7$

These are the coordinates of the center of the circle. These are denoted as $C\left(-5,-7\right)$.

To find the radius of the circle we know the formula is

$\mathit{r}\mathbf{=}\sqrt{{\mathbf{h}}^{\mathbf{2}}\mathbf{+}{\mathbf{k}}^{\mathbf{2}}\mathbf{-}\mathbf{c}}$

Using this formula, the radius will be

$\begin{array}{rcl}\therefore r& =& \sqrt{{\left(-5\right)}^2+{\left(-7\right)}^2-\left(-51\right)}\\ & =& \sqrt{25+49+51}\\ & =& \sqrt{125}\\ & =& 5\sqrt{5}\end{array}$

Step 2: Calculation for the longest and shortest distance from the given point

Let's call the point with coordinates $\left(-7,2\right)$ be $A$.

To find the distance $CA$ we use the distance formula that is

$\mathit{d}\mathbf{=}\sqrt{{\left(x_2+x_1\right)}^{\mathbf{2}}\mathbf{+}{\left(y_2+y_1\right)}^{\mathbf{2}}}$

Here, $x_2=-7,x_1=-5,y_2=2,\text{and}{y}_1=-7$, then

$\begin{array}{rcl}\therefore CA& =& \sqrt{{\left(-7-5\right)}^2+{\left(2-7\right)}^2}\\ & =& \sqrt{{\left(-12\right)}^2+{\left(-5\right)}^2}\\ & =& \sqrt{144+25}\\ & =& \sqrt{169}\\ & =& 13\text{units}\end{array}$

Now, the longest distance will be,

$\begin{array}{rcl}\therefore p& =& CA+r\\ & =& 13+5\sqrt{5}\text{units}\end{array}$

The shortest distance will be,

$\begin{array}{rcl}\therefore q& =& CA-r\\ & =& 13-5\sqrt{5}\text{units}\end{array}$

Step 3: Calculation for the required geometric mean

The formula for the geometric mean between two numbers $a\text{and}b$ is

$\mathbf{\text{GM}}\mathbf{=}\sqrt{\mathbf{a}\mathbf{b}}$

In this case, $a=p\text{and}b=q$, then

$\begin{array}{rcl}\therefore GM& =& \sqrt{pq}\\ & =& \sqrt{\left(13+5\sqrt{5}\right)\left(13-5\sqrt{5}\right)}\\ & =& \sqrt{{\left(13\right)}^2-{\left(5\sqrt{5}\right)}^2}\\ & =& \sqrt{169-125}\\ & =& \sqrt{44}\\ & =& 2\sqrt{11}\end{array}$

Hence, the correct option is (A).