Mathematics Standard 12 Vol I>Chapter -1 - Two Dimensional Analytical Geometry-II>EXERCISE>Q 4

MEDIUM

12th Tamil Nadu Board

IMPORTANT

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An engineer designs a satellite dish with a parabolic cross section. The dish is $5 m$ wide at the opening, and the focus is placed $1.2 m$ from the vertex. Find the depth of the satellite dish at the vertex.

Important Points to Remember in Chapter -1 - Two Dimensional Analytical Geometry-II from Tamil Nadu Board Mathematics Standard 12 Vol I Solutions

1. Introduction of Conic Section:

A conic section or conic is the locus of a point $P$ which moves in such a way that its distances from a fixed point $S$ always bears a constant ratio to its distance from a fixed line, all being in the same plane.

2. Definitions of Various Important Terms

(i) Focus: The fixed point is called the focus of the conic section.

(ii) Directrix: The fixed straight line is called the directrix of the conic section.

(iii) Eccentricity: The constant ratio is called the eccentricity of the conic section and is denoted by $e$ .

(iv) Axis: The straight line passing through the focus and perpendicular to the directrix is called the axis of the conic section.

(v) Vertex: The points of intersection of the conic section and the axis are called vertices of the conic section.

(vi) Centre: The point which bisects every chord of the conic passing through it is called the centre of the conic.

(vii) Latus-Rectum: The latus-rectum of a conic is the chord passing through the focus and perpendicular to the axis.

3. Circles

A circle is the set of all points in a plane that are equidistant from a fixed point in the plane.The fixed point is called the centre of the circle and the distance from the centre to a point on the circle is called the radius of the circle.

(i) Standard Equation of a Circle: The equation of a circle with centre $C (h, k)$ and radius $r$ is given by ${(x - h)}^{2} + {(y - k)}^{2} = r^{2}$ .

The above equation is known as the central form of the equation of a circle.

(ii) The equation of a circle with centre $C (0, 0)$ and radius $r$ is given by $x^{2} + y^{2} = r^{2}$ .

(iii) Some Particular Cases

(a) When the centre of the circle coincides with the origin then equation is $x^{2} + y^{2} = a^{2}$ .

(b) When the circle passes through the origin then equation of the circle will become $x^{2} + y^{2} - 2 h x - 2 k y = 0$ .

(c) When the circle touches $x - a x i s$ then equation is $x^{2} + y^{2} - 2 h x - 2 a y + h^{2} = 0$ .

(d) When the circle touches $y - a x i s$ then equation of circle will become $x^{2} + y^{2} - 2 a x - 2 k y + k^{2} = 0$ .

(e) When the circle touches both the axes the equation is $x^{2} + y^{2} - 2 a x - 2 a y + a^{2} = 0$ .

(f) When the circle passes through the origin and centre lies on $x - a x i s$ then equation of circle will become $x^{2} + y^{2} - 2 a x = 0$

(g) When the circle passes through the origin and centre lies on $y - a x i s$ then equation is $x^{2} + y^{2} - 2 a y = 0$ .

(iv) General Equation of a Circle

The general equation of a circle is $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ . Its centre is $(- g, - f)$ and radius $= \sqrt{g^{2} + f^{2} - c}$ .

(a) If $g^{2} + f^{2} - c = 0$ then the radius of the circle is zero. Such a circle is known as a point circle.

(b) If $(g^{2} + f^{2} - c) < 0$ then the radius $\sqrt{g^{2} + f^{2} - c^{2}}$ of the circle is imaginary but the centre is real. Such a circle is called an imaginary circle as it is not possible to draw such a circle.

(c) If $g^{2} + f^{2} - c^{2} > 0$ then the radius of the circle is real and hence the circle is also real

(v) Diameter form of Circle

The equation of the circle drawn on the straight line joining two given points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ as diameter is $(x - x_{1}) (x - x_{2}) + (y - y_{1}) (y - y_{2}) = 0$ .

4. Parabola

A parabola is the locus of a point which moves in a plane such that its distance from a fixed point in the plane is always equal to its distance from a fixed straight line in the same plane.

Equation of the Parabola in its Standard Form is given by $y^{2} = 4 a x$ .

(i) Some Other Standard Forms of Parabola

	$y^{2} = 4 a x$	$y^{2} = - 4 a x$	$x^{2} = 4 a y$	$y^{2} = - 4 a y$
Coordinates of vertex	$(0, 0)$	$(0, 0)$	$(0, 0)$	$(0, 0)$
Coordinates of focus	$(a, 0)$	$(- a, 0)$	$(0, a)$	$(0, - a)$
Equation of the directrix	$x = - a$	$x = a$	$y = - a$	$y = a$
Equation of the axis	$y = 0$	$y = 0$	$x = 0$	$x = 0$
Length of the Latus-rectum	$4 a$	$4 a$	$4 a$	$4 a$
Focal distance of a point $P (x, y)$	$a + x$	$a - x$	$a + y$	$a - y$

5. Ellipse

An ellipse is the locus of a point in a plane which moves in such a way that the ratio of its distance from a fixed point (called focus) in the same plane to its distance from a fixed straight line (called directrix) is always constant which is always less than unity.

Equation of the Ellipse in the Standard Form is given by $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ .

(i) Some Other Standard Forms of Ellipse

	$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1, a > b$	$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1, a < b$
Coordinates of the centre	$(0, 0)$	$(0, 0)$
Coordinates of the vertices	$(a, 0)$ and $(- a, 0)$	$(0, b)$ and $(0, - b)$
Coordinates of foci	$(a e, 0)$ and $(- a e, 0)$	$(0, b e)$ and $(0, - b e)$
Length of the major axis	$2 a$	$2 b$
Length of the minor axis	$2 b$	$2 a$
Equation of the major axis	$y = 0$	$x = 0$
Equation of the minor axis	$x = 0$	$y = 0$
Equation of the directrices	$x = \frac{a}{e}$ and $x = - \frac{a}{e}$	$y = \frac{b}{e}$ and $y = - \frac{b}{e}$
Eccentricity	$e = \sqrt{1 - \frac{b^{2}}{a^{2}}}$	$e = \sqrt{1 - \frac{a^{2}}{b^{2}}}$
Length of the latus-rectum	$\frac{2 b^{2}}{a}$	$\frac{2 a^{2}}{b}$
Focal distance of a point $(x, y)$	$a \pm e x$	$b \pm e y$

6. Hyperbola

A hyperbola is the locus of a point in a plane which moves in such a way that the ratio of its distance from a fixed point (called focus) in the same plane to its distance from a fixed line (called directrix) is always constant which is always greater than unity.

Equation of the Hyperbola in the Standard Form is given by $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ .

(i) Some Other Standard Forms of Hyperbola

	Hyperbola	Conjugate hyperbola
Coordinates of the centre	$(0, 0)$	$(0, 0)$
Coordinates of the vertices	$(a, 0)$ and $(- a, 0)$	$(0, b)$ and $(0, - b)$
Coordinates of the foci	$(\pm a e, 0)$	$(0, \pm b e)$
Length of the transverse axis	$2 a$	$2 b$
Length of conjugative axis	$2 b$	$2 a$
Equation of the directrices	$x = \pm \frac{a}{e}$	$y = \pm \frac{b}{e}$
Eccentricity	$e = \sqrt{\frac{a^{2} + b^{2}}{a^{2}}}$ or $b^{2} = a^{2} (e^{2} - 1)$	$e = \sqrt{\frac{b^{2} + a^{2}}{b^{2}}}$ or $a^{2} = b^{2} (e^{2} - 1)$
Length of the latus-rectum	$\frac{2 b^{2}}{a}$	$\frac{2 a^{2}}{b}$
Equation of the transverse axis	$y = 0$	$x = 0$
Equation of the conjugate axis	$x = 0$	$y = 0$

7. Tangents and normals to conics:

(i) Tangents and normals to parabola:

(a) Point of contact of the tangent $y y_{1} = 2 a (x + x_{1})$ with the parabola $y^{2} = 4 a x$ is given by $(x, x_{1})$ .(point form)

(b) Point of contact of the tangent $y = m x + \frac{a}{m}$ with the parabola $y^{2} = 4 a x$ is given by $(\frac{a}{m}, \frac{2 a}{m})$ . (slope form)

(c) Point of contact of the tangent $t y = x + a t^{2}$ with the parabola $y^{2} = 4 a x$ is given by $(a t^{2}, 2 a t)$ . (parametric form)

(d) The equation of the tangent of the parabola $y^{2} = 4 a x$ at $(a t^{2}, 2 a t)$ is given by $y = x + a t^{2}$ .

(e) The equation of the tangent to the parabola $y^{2} = 4 a x$ at a point $(x_{1}, y_{1})$ is given by $y y_{1} = 2 a (x + x_{1})$ .

(f) The equation of normal to the parabola $y^{2} = 4 a x$ at a point $(x_{1}, y_{1})$ is given by $(y - y_{1}) = - \frac{y_{1}}{2 a} (x - x_{1})$ .

(g) Equation of normal to the parabola $y^{2} = 4 a x$ at the point $(a m^{2}, - 2 a m)$ is given by $y = m x - 2 a m - a m^{3}$ .

(ii) Tangents and normals to ellipse:

(a) The equation of tangent to the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ at $(x_{1}, y_{1})$ is $\frac{x x_{1}}{a^{2}} + \frac{y y_{1}}{b^{2}} = 1$ .

(b) The equation of normal to the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ at $(x_{1}, y_{1})$ is $a^{2} y_{1} (x - x_{1}) = b^{2} x_{1} (y - y_{1})$ .

(c) The equation of tangent to the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ at $(a \cos θ, b \sin θ)$ is $b x \cos θ + a y \sin θ - a b = 0$ .

(d) The equation of normal to the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ at $(a cosθ, b sinθ)$ is $axsec θ - b y cosec θ = a^{2} - b^{2}$ .

(iii) Tangents and normals to hyperbola:

(a) The equations of the tangent and normal to the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ at the point $(x_{1}, y_{1})$ are $\frac{x x_{1}}{a^{2}} - \frac{y y_{1}}{b^{2}} = 1$ and $a^{2} y_{1} x + b^{2} x_{1} y - (a^{2} + b^{2}) x_{1} y_{1} = 0$ respectively.

(b) The equation of tangent to the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ at $(a secθ, b tanθ)$ is $b x secθ - a y tanθ - a b = 0$ .

(c) The equation of normal to the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ at $(a \sec θ, b tanθ)$ is $a x cosθ + b y cotθ = a^{2} + b^{2}$ .