# Find the shortest and largest distance from the point $\left( 2,-7 \right)$ to the circle

${{x}^{2}}+{{y}^{2}}-14x-10y-151=0$

Answer

Verified

363.6k+ views

Hint: Find the distance of the point from the centre of the circle. After doing so, add the radius to it for the largest distance, and subtract the radius from it for shortest distance.

Complete step-by-step answer:

As, the general equation of the circle is ${{\left( x-h \right)}^{2}}+{{\left( y-k \right)}^{2}}={{r}^{2}}$ where $(h,k)$ is the centre and $r$ is the radius.

Now the given equation is : ${{x}^{2}}+{{y}^{2}}-14x-10y-151=0$

Adding $49+25$ to both sides, we get :

$\begin{align}

& \Rightarrow {{x}^{2}}-14x+49+{{y}^{2}}-10y+25=151+49+25 \\

& \Rightarrow {{\left( x-7 \right)}^{2}}+{{\left( y-5 \right)}^{2}}=225 \\

& \Rightarrow {{\left( x-7 \right)}^{2}}+{{\left( y-5 \right)}^{2}}={{\left( 15 \right)}^{2}} \\

\end{align}$

So, the circle has its centre at $\left( 7,5 \right)$ and has a radius $r=15$ units.

Then, the shortest distance between the point $({{x}_{1}},{{y}_{1}})$ and the circle is given by the distance of that point from the centre of the circle minus the radius of the circle.

Hence, if $d$ is the shortest distance of the point $({{x}_{1}},{{y}_{1}})$ from the circle, then :

\[\begin{align}

& d=\left| \sqrt{{{\left( {{x}_{1}}-h \right)}^{2}}+{{\left( {{y}_{1}}-k \right)}^{2}}}-r \right| \\

& =\left| \sqrt{{{\left( 2-7 \right)}^{2}}+{{\left( -7-5 \right)}^{2}}}-15 \right| \\

& =\left| \sqrt{{{\left( -5 \right)}^{2}}+{{\left( -12 \right)}^{2}}}-15 \right| \\

& =\left| \sqrt{169}-15 \right|=|13-15|=2 \\

\end{align}\]

Therefore, the shortest distance of point with coordinates $({{x}_{1}},{{y}_{1}})$ from the circle$=d=2\text{ Unit }$

Now, the longest distance between the point $({{x}_{1}},{{y}_{1}})$ and the circle is defined as the sum of the distance of point $({{x}_{1}},{{y}_{1}})$ from the centre of the circle, and the circle’s radius.

Hence, if $D$ is the largest distance of the point $({{x}_{1}},{{y}_{1}})$ from the circle, then :

\[\begin{align}

& D=\left| \sqrt{{{\left( {{x}_{1}}-h \right)}^{2}}+{{\left( {{y}_{1}}-k \right)}^{2}}}+r \right| \\

& =\left| \sqrt{{{\left( -5 \right)}^{2}}+{{\left( -12 \right)}^{2}}}+15 \right| \\

& =\left| \sqrt{169}+15 \right|=|13+15|=28 \\

\end{align}\]

Therefore, the longest distance of the point $({{x}_{1}},{{y}_{1}})$ from the circle $=D=28\text{ units}\text{.}$

Note: The most common mistake students make is in finding points and remembering formulae.

Remember that, \[D=\left| \sqrt{{{\left( {{x}_{1}}-h \right)}^{2}}+{{\left( {{y}_{1}}-k \right)}^{2}}}+r \right|\text{ }\]always, \[d=\left| \sqrt{{{\left( {{x}_{1}}-h \right)}^{2}}+{{\left( {{y}_{1}}-k \right)}^{2}}}-r \right|\text{ }\]always. There might also be some confusion in finding the centre point and the radius. So, you can just find the compare the equation given with the general equation of a circle which is ${{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0$, where the circle’s centre $=(-g,-f)$, and its radius $=\sqrt{{{g}^{2}}+{{f}^{2}}-c}$. Also, be very cautious and avoid calculation mistakes.

Complete step-by-step answer:

As, the general equation of the circle is ${{\left( x-h \right)}^{2}}+{{\left( y-k \right)}^{2}}={{r}^{2}}$ where $(h,k)$ is the centre and $r$ is the radius.

Now the given equation is : ${{x}^{2}}+{{y}^{2}}-14x-10y-151=0$

Adding $49+25$ to both sides, we get :

$\begin{align}

& \Rightarrow {{x}^{2}}-14x+49+{{y}^{2}}-10y+25=151+49+25 \\

& \Rightarrow {{\left( x-7 \right)}^{2}}+{{\left( y-5 \right)}^{2}}=225 \\

& \Rightarrow {{\left( x-7 \right)}^{2}}+{{\left( y-5 \right)}^{2}}={{\left( 15 \right)}^{2}} \\

\end{align}$

So, the circle has its centre at $\left( 7,5 \right)$ and has a radius $r=15$ units.

Then, the shortest distance between the point $({{x}_{1}},{{y}_{1}})$ and the circle is given by the distance of that point from the centre of the circle minus the radius of the circle.

Hence, if $d$ is the shortest distance of the point $({{x}_{1}},{{y}_{1}})$ from the circle, then :

\[\begin{align}

& d=\left| \sqrt{{{\left( {{x}_{1}}-h \right)}^{2}}+{{\left( {{y}_{1}}-k \right)}^{2}}}-r \right| \\

& =\left| \sqrt{{{\left( 2-7 \right)}^{2}}+{{\left( -7-5 \right)}^{2}}}-15 \right| \\

& =\left| \sqrt{{{\left( -5 \right)}^{2}}+{{\left( -12 \right)}^{2}}}-15 \right| \\

& =\left| \sqrt{169}-15 \right|=|13-15|=2 \\

\end{align}\]

Therefore, the shortest distance of point with coordinates $({{x}_{1}},{{y}_{1}})$ from the circle$=d=2\text{ Unit }$

Now, the longest distance between the point $({{x}_{1}},{{y}_{1}})$ and the circle is defined as the sum of the distance of point $({{x}_{1}},{{y}_{1}})$ from the centre of the circle, and the circle’s radius.

Hence, if $D$ is the largest distance of the point $({{x}_{1}},{{y}_{1}})$ from the circle, then :

\[\begin{align}

& D=\left| \sqrt{{{\left( {{x}_{1}}-h \right)}^{2}}+{{\left( {{y}_{1}}-k \right)}^{2}}}+r \right| \\

& =\left| \sqrt{{{\left( -5 \right)}^{2}}+{{\left( -12 \right)}^{2}}}+15 \right| \\

& =\left| \sqrt{169}+15 \right|=|13+15|=28 \\

\end{align}\]

Therefore, the longest distance of the point $({{x}_{1}},{{y}_{1}})$ from the circle $=D=28\text{ units}\text{.}$

Note: The most common mistake students make is in finding points and remembering formulae.

Remember that, \[D=\left| \sqrt{{{\left( {{x}_{1}}-h \right)}^{2}}+{{\left( {{y}_{1}}-k \right)}^{2}}}+r \right|\text{ }\]always, \[d=\left| \sqrt{{{\left( {{x}_{1}}-h \right)}^{2}}+{{\left( {{y}_{1}}-k \right)}^{2}}}-r \right|\text{ }\]always. There might also be some confusion in finding the centre point and the radius. So, you can just find the compare the equation given with the general equation of a circle which is ${{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0$, where the circle’s centre $=(-g,-f)$, and its radius $=\sqrt{{{g}^{2}}+{{f}^{2}}-c}$. Also, be very cautious and avoid calculation mistakes.

Last updated date: 30th Sep 2023

•

Total views: 363.6k

•

Views today: 8.63k

Recently Updated Pages

What is the Full Form of DNA and RNA

What are the Difference Between Acute and Chronic Disease

Difference Between Communicable and Non-Communicable

What is Nutrition Explain Diff Type of Nutrition ?

What is the Function of Digestive Enzymes

What is the Full Form of 1.DPT 2.DDT 3.BCG

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell

Why are resources distributed unequally over the e class 7 social science CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Briefly mention the contribution of TH Morgan in g class 12 biology CBSE

What is the past tense of read class 10 english CBSE