Length of y - intercept made by the circle $5{{\text{x}}^2} + 5{{\text{y}}^2} - 2{\text{x + 6y - 8 = 0 is:}}$
$
{\text{A}}{\text{.}}\dfrac{{19}}{5}. \\
{\text{B}}{\text{.}}\dfrac{{14}}{5}. \\
{\text{C}}{\text{.}}\dfrac{{11}}{5}. \\
{\text{D}}{\text{.}}\dfrac{9}{5}. \\
$
Answer
381.9k+ views
Hint: In this question, the equation of circle is given. To find the y-intercept made by a circle we will first convert the given circle equation into the standard circle equation and then use the formula for y-intercept to the value of y-intercept.
Complete step-by-step answer:
In the question, it is given that:
Equation of circle is $5{{\text{x}}^2} + 5{{\text{y}}^2} - 2{\text{x + 6y - 8 = 0}}$ .
We have to find the y-intercept made by the given circle.
We know that the standard equation of circle is given by:
${{\text{x}}^2} + {{\text{y}}^2} + 2{\text{gx + 2fy + c = 0}}$. (1)
Y-intercept of this circle is given by $2\sqrt {{{\text{f}}^2} - c} $ .
But the equation of the circle given is not in the standard form. So we will first convert this equation in standard form.
$
\Rightarrow 5{{\text{x}}^2} + 5{{\text{y}}^2} - 2{\text{x + 6y - 8 = 0}} \\
$
On dividing the above equation by 5, we get,
$
\Rightarrow {{\text{x}}^2} + {{\text{y}}^2} - \dfrac{2}{5}{\text{x + }}\dfrac{6}{5}{\text{y - }}\dfrac{8}{5}{\text{ = 0}} \\
$
So the above equation is the standard equation of a circle.
On comparing the above equation with equation 1, we get:
$
2{\text{g = - }}\dfrac{2}{5} \\
\Rightarrow {\text{g = - }}\dfrac{1}{5}. \\
{\text{And}} \\
{\text{2f = }}\dfrac{6}{5}. \\
\Rightarrow {\text{f = }}\dfrac{3}{5},{\text{ and c = - }}\dfrac{8}{5}. \\
$
Y-intercept made by circle =$2\sqrt {{{\text{f}}^2} - c} $.
Putting the value of ‘f’ and ‘c’ in the above formula, we get:
Y-intercept made by circle = $2\sqrt {{{\text{f}}^2} - c} = 2\sqrt {{{\left( {\dfrac{3}{5}} \right)}^2} - \left( { - \dfrac{8}{5}} \right)} = 2\sqrt {\dfrac{9}{{25}} + \dfrac{8}{5}} = 2\sqrt {\dfrac{{49}}{{25}}} = 2 \times \dfrac{7}{5} = \dfrac{{14}}{5}.$
Note: In this type of question, the first important thing is to clearly see the question whether it is asking y-intercept or x-intercept. Then convert the given equation into a standard form of equation. You should remember the formula for finding y-intercept. Compare the transformed given equation with the standard equation to get the value of parameters required for computing the y-intercept.
Complete step-by-step answer:
In the question, it is given that:
Equation of circle is $5{{\text{x}}^2} + 5{{\text{y}}^2} - 2{\text{x + 6y - 8 = 0}}$ .
We have to find the y-intercept made by the given circle.
We know that the standard equation of circle is given by:
${{\text{x}}^2} + {{\text{y}}^2} + 2{\text{gx + 2fy + c = 0}}$. (1)
Y-intercept of this circle is given by $2\sqrt {{{\text{f}}^2} - c} $ .
But the equation of the circle given is not in the standard form. So we will first convert this equation in standard form.
$
\Rightarrow 5{{\text{x}}^2} + 5{{\text{y}}^2} - 2{\text{x + 6y - 8 = 0}} \\
$
On dividing the above equation by 5, we get,
$
\Rightarrow {{\text{x}}^2} + {{\text{y}}^2} - \dfrac{2}{5}{\text{x + }}\dfrac{6}{5}{\text{y - }}\dfrac{8}{5}{\text{ = 0}} \\
$
So the above equation is the standard equation of a circle.
On comparing the above equation with equation 1, we get:
$
2{\text{g = - }}\dfrac{2}{5} \\
\Rightarrow {\text{g = - }}\dfrac{1}{5}. \\
{\text{And}} \\
{\text{2f = }}\dfrac{6}{5}. \\
\Rightarrow {\text{f = }}\dfrac{3}{5},{\text{ and c = - }}\dfrac{8}{5}. \\
$
Y-intercept made by circle =$2\sqrt {{{\text{f}}^2} - c} $.
Putting the value of ‘f’ and ‘c’ in the above formula, we get:
Y-intercept made by circle = $2\sqrt {{{\text{f}}^2} - c} = 2\sqrt {{{\left( {\dfrac{3}{5}} \right)}^2} - \left( { - \dfrac{8}{5}} \right)} = 2\sqrt {\dfrac{9}{{25}} + \dfrac{8}{5}} = 2\sqrt {\dfrac{{49}}{{25}}} = 2 \times \dfrac{7}{5} = \dfrac{{14}}{5}.$
Note: In this type of question, the first important thing is to clearly see the question whether it is asking y-intercept or x-intercept. Then convert the given equation into a standard form of equation. You should remember the formula for finding y-intercept. Compare the transformed given equation with the standard equation to get the value of parameters required for computing the y-intercept.
Recently Updated Pages
Define absolute refractive index of a medium

Find out what do the algal bloom and redtides sign class 10 biology CBSE

Prove that the function fleft x right xn is continuous class 12 maths CBSE

Find the values of other five trigonometric functions class 10 maths CBSE

Find the values of other five trigonometric ratios class 10 maths CBSE

Find the values of other five trigonometric functions class 10 maths CBSE

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

Which one of the following places is unlikely to be class 8 physics CBSE

Select the word that is correctly spelled a Twelveth class 10 english CBSE

Difference Between Plant Cell and Animal Cell

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

What is the past tense of read class 10 english CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Elucidate the structure of fructose class 12 chemistry CBSE

What is pollution? How many types of pollution? Define it
