Answer

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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.

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