Answer
Verified
479.7k+ views
Hint: Find the points of intersection of the line and circle, to find the x-coordinate or abscissa. Then find the geometric means using the formula$\sqrt{ab}$.
Complete step-by-step answer:
In the given question, the equation of the circle is,
${{x}^{2}}+{{y}^{2}}-4x-6y+7=0$
The equation of line is,
y = 1
We can find the point of intersection of line and the circle by putting ‘y = 1’ in equation of the circle,
${{x}^{2}}+{{y}^{2}}-4x-6y+7=0$
By substituting, we get
${{x}^{2}}+{{(1)}^{2}}-4x-6(1)+7=0$
$\Rightarrow {{x}^{2}}-4x+2=0$
Now we will convert this into square,
$\Rightarrow {{x}^{2}}-4x+4=2$
$\Rightarrow {{x}^{2}}-2x-2x+4=2$
$\Rightarrow {{\left( x-2 \right)}^{2}}=2$
Taking square on both sides, we get
\[\Rightarrow x-2=\pm \sqrt{2}\]
\[\therefore x=2\pm \sqrt{2}\]
Abscissa represents the value of x-coordinate corresponding to that point.
So, the abscissa of points which we found are $\left( 2+\sqrt{2} \right)$ and $\left( 2-\sqrt{2} \right)$.
In the question we are asked to find the geometric mean of the values.
We know the formula that the geometric mean between two values a,b is $\sqrt{ab}$.
So, geometric mean in this case will be,
$\sqrt{\left( 2+\sqrt{2} \right)\left( 2-\sqrt{2} \right)}$
Now we know the formula, $(a+b)(a-b)={{a}^{2}}-{{b}^{2}}$ , so the above equation can be written as,
$\sqrt{{{2}^{2}}-{{\left( \sqrt{2} \right)}^{2}}}=\sqrt{4-2}=\sqrt{2}$
So, the geometric means of the abscissa of the points of intersection of the circle ${{x}^{2}}+{{y}^{2}}-4x-6y+7=0$ and the line $y=1$ is $\sqrt{2}$ .
Hence, the correct answer is option (b).
Note: In this question, students should be careful while working out and finding the values of x.
Common mistakes made by students are while writing the formula of geometric mean. They find the abscissa and get to stop there itself.
Complete step-by-step answer:
In the given question, the equation of the circle is,
${{x}^{2}}+{{y}^{2}}-4x-6y+7=0$
The equation of line is,
y = 1
We can find the point of intersection of line and the circle by putting ‘y = 1’ in equation of the circle,
${{x}^{2}}+{{y}^{2}}-4x-6y+7=0$
By substituting, we get
${{x}^{2}}+{{(1)}^{2}}-4x-6(1)+7=0$
$\Rightarrow {{x}^{2}}-4x+2=0$
Now we will convert this into square,
$\Rightarrow {{x}^{2}}-4x+4=2$
$\Rightarrow {{x}^{2}}-2x-2x+4=2$
$\Rightarrow {{\left( x-2 \right)}^{2}}=2$
Taking square on both sides, we get
\[\Rightarrow x-2=\pm \sqrt{2}\]
\[\therefore x=2\pm \sqrt{2}\]
Abscissa represents the value of x-coordinate corresponding to that point.
So, the abscissa of points which we found are $\left( 2+\sqrt{2} \right)$ and $\left( 2-\sqrt{2} \right)$.
In the question we are asked to find the geometric mean of the values.
We know the formula that the geometric mean between two values a,b is $\sqrt{ab}$.
So, geometric mean in this case will be,
$\sqrt{\left( 2+\sqrt{2} \right)\left( 2-\sqrt{2} \right)}$
Now we know the formula, $(a+b)(a-b)={{a}^{2}}-{{b}^{2}}$ , so the above equation can be written as,
$\sqrt{{{2}^{2}}-{{\left( \sqrt{2} \right)}^{2}}}=\sqrt{4-2}=\sqrt{2}$
So, the geometric means of the abscissa of the points of intersection of the circle ${{x}^{2}}+{{y}^{2}}-4x-6y+7=0$ and the line $y=1$ is $\sqrt{2}$ .
Hence, the correct answer is option (b).
Note: In this question, students should be careful while working out and finding the values of x.
Common mistakes made by students are while writing the formula of geometric mean. They find the abscissa and get to stop there itself.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE
Which are the Top 10 Largest Countries of the World?
How do you graph the function fx 4x class 9 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE