Question

The equation of the circle which touches the x-axis at (1,0) and passes through (0,1) is
(1) x²+y²+2x+2y+1=0
(2) x²+y²-2x-2y+1=0
(3)x²+y²+2x+2y-1=0
(4)x²+y²-2x-2y-1=0​

Answer 1

EXPLANATION.

Equation of the circle which touches x -axis at (1,0).

Passes through point = (0,1).

As we know that,

Let we assume that,

Centre of circle = (1,k).

As we know that,

General equation of circle.

⇒ (x - h)² + (y - k)² = k².

We can write equation as,

⇒ (x - 1)² + (y - k)² = k².

It passes through point = (0,1).

It means (0,1) satisfy the equation.

⇒ (0 - 1)² + (1 - k)² = k².

⇒ (-1)² + (1 - k)² = k².

⇒ 1 + 1 + k² - 2k = k².

⇒ 2 - 2k = 0.

⇒ k = 1.

Put the values of k = 1 in the equation, we get.

⇒ (x - 1)² + (y - k)² = k².

⇒ (x - 1)² + (y - 1)² = (1)².

⇒ x² + 1 - 2x + y² + 1 - 2y = 1.

⇒ x² + y² - 2x - 2y + 1 = 0.

Option [B] is correct answer.

                                                                                                                 

MORE INFORMATION.

Intercepts made by a circle on the axes.

The intercept made by the circle x² + y² + 2gx + 2fy + c = 0 on the co-ordinate axes are : 2√g² - c  and  2√f²- c.

(1) g² - c > 0 : Circle cuts the x axis at two distinct points.

(2) g² = c : Circle touches the x axis.

(3) g² < c : Circle lies completely above or below the x axis.

Answer 2

$\huge\red{A}\pink{N}\orange{S} \green{W}\blue{E}\gray{R} =$

EXPLANATION.

Equation of the circle which touches x

-axis at (1.0).

Passes through point = (0.1).

As we know that.

Let we assume that, Centre of circle = (1.k).

As we know that,

General equation of circle.

→ (x-h)² + (y-k)² = k².

We can write equation as,

- (x - 1)² + (y-k)² = k². It passes through point = (0,1). It means (0.1) satisfy the equation.

(0-1)²+(1-k)²=k².

- (-1)² + (1-k)² =k². >1+1+k²-2k=k².

→ 2-2k = 0.

k=1.

Put the values of k=1 in the equation, we get.

- (x-1)² + (y-k)² = k².

(x - 1)² + (x - 1)² = (1²

x²+1-2x + y² +1-2y = 1. = x² + y²-2x-2y+1=0.

Option [B] is correct answer,

MORE INFORMATION.

Intercepts made by a circle on the axes.

The intercept made by the circle x² + y² + 2gx+2y+c=0 on the co-ordinate axes are:2vg²-c and 2-vf-c.

(1) g-c>0: Circle cuts the x axis at two distinct points

(2) g²=c: Circle touches the x axis. (3) g² <c: Circle lies completely above or below the x axis.