Math Problems Vault

Problem 7.1

(1)(2)(3)(4)

Find the radius and the coordinates of the center of the circle:

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(1)

$x^{2} + y^{2} + 4y = 0$

(2)

$x^{2} + y^{2} + 5x - 5y + 12 = 0$

(3)

$2x^{2} + 2y^{2} - 12x + y + 3 = 0$

(4)

$7x^{2} + 7y^{2} - 2x - 7y - 1 = 0$ .

Problem 7.2

Under what necessary and sufficient condition does the equation $Ax^{2} + By^{2} + 2Cx + 2Dy + E = 0$ define a circle? Express the radius and the coordinates of the center of the circle in terms of the coefficients of the equation.

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Problem 7.3

Find the equation of the circle with center at the point $M(2, 2)$ , tangent to the line $3x + y - 18 = 0$ .

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Problem 7.4

(1)(2)(3)

Find the necessary and sufficient condition for the line $Ax + By + C = 0$ :

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(1)

to have no common points with the circle $(x - a)^{2} + (y - b)^{2} = R^{2}$

(2)

to have two common points with this circle

(3)

to be tangent to this circle.

Problem 7.5

(1)(2)

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(1)

Find the equation of the tangent line drawn to the circle $(x - 1)^{2} + (y + 2)^{2} = 25$ at the point $M(- 3, 1)$ .

(2)

Find the equations of the tangent lines to the circle $(x - 1)^{2} + (y + 1)^{2} = 9$ passing through the point $M(1, 4)$ .

Problem 7.6

Find the equations of the tangent lines to the circle $(x + 3)^{2} + (y + 1)^{2} = 4$ , parallel to the line $5x - 12y + 1 = 0$ .

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Problem 7.7

(1)(2)

Verify that the two given circles are tangent to each other, and find the equation of their common tangent line passing through the point of tangency:

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(1)

$(x - 1)^{2} + (y - 2)^{2} = 18, (x - 5)^{2} + (y - 6)^{2} = 2$

(2)

$(x + 1)^{2} + (y - 1)^{2} = 45, (x - 1)^{2} + (y - 5)^{2} = 5$ .

Problem 7.8

Find the equations of the common tangent lines to the circles $(x - 2)^{2} + (y + 1)^{2} = 9$ and $(x - 4)^{2} + (y - 3)^{2} = 1$ .

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Problem 7.9

Through a point $A$ lying outside a circle, a line tangent to the circle at a point $B$ is drawn, and another line intersecting the circle at points $C$ and $D$ . Prove that $\left|AB\right|^{2} = \left|AD\right| \cdot \left|AC\right|$

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Problem 7.10

Two circles are externally tangent. Through their point of tangency a line is drawn, meeting the first circle at one more point $A$ and the second circle at one more point $B$ . Prove that the tangent lines to the circles at the points $A$ and $B$ are parallel.

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