Answer
Verified
414.9k+ views
Hint: First we find the centre of the given circle and then find the distance of point from the line x-2y-1 =0. If the distance comes out to be equal to the radius of the circle then the given line is tangent to the circle.
Complete step-by-step answer:
The general equation of a circle is given as:
${{\text{x}}^2} + {{\text{y}}^2}{\text{ + 2gx + 2fy + c = 0 }}$ (1)
The centre of this circle is given as: O (-g ,-f)
The given equation of circle is ${{\text{x}}^2} + {{\text{y}}^2} - 8{\text{x + 2y + 12 = 0 }}$
Comparing this equation with equation 1, we get:
2g = 8
$ \Rightarrow $ g = 4 and
2f = 2
$ \Rightarrow $f =1
$\therefore $ Centre O (-4 , -1)
We know that length of perpendicular drawn from point(${{\text{x}}_1},{{\text{y}}_1}$) is given by:
d = $\dfrac{{{\text{|a}}{{\text{x}}_1} + {\text{b}}{{\text{y}}_1} + c|}}{{\sqrt {{{\text{a}}^2} + {{\text{b}}^2}} }}$
Putting the values in above equation, we get:
d = $\dfrac{{{\text{| - 4}} + ( - 2 \times - 4) - 1|}}{{\sqrt {{1^2} + {2^2}} }} = \sqrt 5 $
Now, we calculate the radius of the circle.
r= $\sqrt {{{\text{g}}^2} + {{\text{f}}^2} - c} = \sqrt {{4^2} + {1^2} - 12} = \sqrt 5 $
$\because $ The distance from centre of circle to the given line is equal to the radius of the radius.
So, the given line is tangent to the given circle.
Note- In such a type of question, you should know how to check for tangency of a line to a circle. For a circle ${{\text{x}}^2} + {{\text{y}}^2} = {r^2}$ and a line y = mx +c, the condition of tangency is given as:
${{\text{c}}^2} = {r^2}(1 + {m^2})$ . And if general equation of the circle is given i.e${{\text{x}}^2} + {{\text{y}}^2}{\text{ + 2gx + 2fy + c = 0 }}$then in this case calculate distance from (-g,-f) to the line and it must be equal to radius of circle.
Complete step-by-step answer:
The general equation of a circle is given as:
${{\text{x}}^2} + {{\text{y}}^2}{\text{ + 2gx + 2fy + c = 0 }}$ (1)
The centre of this circle is given as: O (-g ,-f)
The given equation of circle is ${{\text{x}}^2} + {{\text{y}}^2} - 8{\text{x + 2y + 12 = 0 }}$
Comparing this equation with equation 1, we get:
2g = 8
$ \Rightarrow $ g = 4 and
2f = 2
$ \Rightarrow $f =1
$\therefore $ Centre O (-4 , -1)
We know that length of perpendicular drawn from point(${{\text{x}}_1},{{\text{y}}_1}$) is given by:
d = $\dfrac{{{\text{|a}}{{\text{x}}_1} + {\text{b}}{{\text{y}}_1} + c|}}{{\sqrt {{{\text{a}}^2} + {{\text{b}}^2}} }}$
Putting the values in above equation, we get:
d = $\dfrac{{{\text{| - 4}} + ( - 2 \times - 4) - 1|}}{{\sqrt {{1^2} + {2^2}} }} = \sqrt 5 $
Now, we calculate the radius of the circle.
r= $\sqrt {{{\text{g}}^2} + {{\text{f}}^2} - c} = \sqrt {{4^2} + {1^2} - 12} = \sqrt 5 $
$\because $ The distance from centre of circle to the given line is equal to the radius of the radius.
So, the given line is tangent to the given circle.
Note- In such a type of question, you should know how to check for tangency of a line to a circle. For a circle ${{\text{x}}^2} + {{\text{y}}^2} = {r^2}$ and a line y = mx +c, the condition of tangency is given as:
${{\text{c}}^2} = {r^2}(1 + {m^2})$ . And if general equation of the circle is given i.e${{\text{x}}^2} + {{\text{y}}^2}{\text{ + 2gx + 2fy + c = 0 }}$then in this case calculate distance from (-g,-f) to the line and it must be equal to radius of circle.
Recently Updated Pages
Basicity of sulphurous acid and sulphuric acid are
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
What is the stopping potential when the metal with class 12 physics JEE_Main
The momentum of a photon is 2 times 10 16gm cmsec Its class 12 physics JEE_Main
Using the following information to help you answer class 12 chemistry CBSE
Trending doubts
State the differences between manure and fertilize class 8 biology CBSE
Why are xylem and phloem called complex tissues aBoth class 11 biology CBSE
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Difference Between Plant Cell and Animal Cell
What would happen if plasma membrane ruptures or breaks class 11 biology CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
What precautions do you take while observing the nucleus class 11 biology CBSE
What would happen to the life of a cell if there was class 11 biology CBSE
Change the following sentences into negative and interrogative class 10 english CBSE