Coordinate Geometry

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What is Coordinate Geometry?

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A branch of mathematics that uses algebraic equations to describe geometric properties and relationships.

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What is Coordinate Geometry?

A branch of mathematics that uses algebraic equations to describe geometric properties and relationships.

What is the Cartesian coordinate system?

A two-dimensional plane defined by two perpendicular axes: the x-axis and the y-axis.

How is a point represented in the Cartesian plane?

By an ordered pair $(x, y)$, where $x$ is the horizontal distance and $y$ is the vertical distance from the origin.

What is the Distance Formula?

The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $$ d = \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$.

What does the Section Formula calculate?

It determines the coordinates of a point dividing a line segment in a given ratio.

What is the formula for internal division in the Section Formula?

$$ (x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) $$.

What is the Slope of a Line?

The slope is defined as the ratio of the vertical change to the horizontal change between two points: $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$.

What is the Slope-Intercept Form of a line?

$$ y = mx + c $$ where $m$ is the slope and $c$ is the y-intercept.

What is the formula for the angle between two lines?

$$ \tan \theta = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right| $$.

What is the standard form of the equation of a circle?

$$ (x - h)^2 + (y - k)^2 = r^2 $$ where $(h, k)$ is the center and $r$ is the radius.

What is the general form of a circle's equation?

$$ x^2 + y^2 + 2gx + 2fy + c = 0 $$.

What defines a parabola?

A set of points equidistant from a fixed point (focus) and a fixed line (directrix).

What is the standard form of a vertical parabola?

$$ (x - h)^2 = 4a(y - k) $$.

What is the standard form of a horizontal ellipse?

$$ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 $$.

What is the formula for the foci of an ellipse?

Foci are located at $(\pm c, 0)$ where $c = \sqrt{a^2 - b^2}$.

What is the standard form of a hyperbola?

For a horizontal hyperbola: $$ \frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1 $$.

What are the asymptotes of a hyperbola?

For the hyperbola $$ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $$, the asymptotes are given by $y = \pm \frac{b}{a}x$.

What is the importance of Coordinate Geometry in mathematics?

It bridges algebra and geometry, allowing for the analysis and solution of complex geometric problems using algebraic methods.

1 / 18

What is Coordinate Geometry?

Flip

Spacebar

A branch of mathematics that uses algebraic equations to describe geometric properties and relationships.

Flip

Spacebar

What is Coordinate Geometry?

A branch of mathematics that uses algebraic equations to describe geometric properties and relationships.

What is the Cartesian coordinate system?

A two-dimensional plane defined by two perpendicular axes: the x-axis and the y-axis.

How is a point represented in the Cartesian plane?

By an ordered pair $(x, y)$, where $x$ is the horizontal distance and $y$ is the vertical distance from the origin.

What is the Distance Formula?

The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $$ d = \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$.

What does the Section Formula calculate?

It determines the coordinates of a point dividing a line segment in a given ratio.

What is the formula for internal division in the Section Formula?

$$ (x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) $$.

What is the Slope of a Line?

The slope is defined as the ratio of the vertical change to the horizontal change between two points: $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$.

What is the Slope-Intercept Form of a line?

$$ y = mx + c $$ where $m$ is the slope and $c$ is the y-intercept.

What is the formula for the angle between two lines?

$$ \tan \theta = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right| $$.

What is the standard form of the equation of a circle?

$$ (x - h)^2 + (y - k)^2 = r^2 $$ where $(h, k)$ is the center and $r$ is the radius.

What is the general form of a circle's equation?

$$ x^2 + y^2 + 2gx + 2fy + c = 0 $$.

What defines a parabola?

A set of points equidistant from a fixed point (focus) and a fixed line (directrix).

What is the standard form of a vertical parabola?

$$ (x - h)^2 = 4a(y - k) $$.

What is the standard form of a horizontal ellipse?

$$ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 $$.

What is the formula for the foci of an ellipse?

Foci are located at $(\pm c, 0)$ where $c = \sqrt{a^2 - b^2}$.

What is the standard form of a hyperbola?

For a horizontal hyperbola: $$ \frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1 $$.

What are the asymptotes of a hyperbola?

For the hyperbola $$ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $$, the asymptotes are given by $y = \pm \frac{b}{a}x$.

What is the importance of Coordinate Geometry in mathematics?

It bridges algebra and geometry, allowing for the analysis and solution of complex geometric problems using algebraic methods.