Syllabus:
General equation of conics – Classification of conics – Standard equations of parabola – Vertex, focus, axis, directrix, focal distance, focal chord, latus-rectum of parabola , Standard equations of ellipse – Vertices, foci, major axis, minor axis, directrices, eccentricity, centre and latus-rectums of ellipse – Simple problems.
Conic: A conic is defined as the locus of a point which moves such that its distance from a fixed point is always ‘e’ times its distance from a fixed straight line.
Focus: The fixed point is called the focus of the conic.
Directrix: The fixed straight line is called the directrix of the conic.
Eccentricity: The constant ratio is called the eccentricity of the conic.
S’ denotes Focus
Line XM denotes Directrix
SP / PM = e
Note:
(i) If e< 1, the conic is called an ellipse.
(ii) If e = 1, the conic is called a parabola.
(iii) If e>1, the conic is called a hyperbola.
General equation of a conic \[Let\ the\ focus\ be\ S(x_1,\ y_1)\ and\ directrix\ be\ the\ line\ a\ x\ +\ b\ y\ +\ c\ =\ 0\ \hspace{10cm}\]
\[P(x,\ y)\ be\ any\ point\ on\ it\ \hspace{18cm}\]
\[SP\ =\ \sqrt{(x\ -\ x_1)^2\ +\ (y\ -\ y_1)^2}\ \hspace{18cm}\]
\[PM\ =\ Perpendicular\ distance\ of\ P\ from\ the\ line\ a\ x\ +\ b\ y\ +\ c\ =\ 0\ \hspace{10cm}\]
\[=\ \pm\ \frac{a\ x\ +\ b\ y\ +\ c}{\sqrt{a^2\ +\ b^2}}\ \hspace{15cm}\]
\[Always\ \frac{SP}{PM}\ =\ e\ \hspace{10cm}\]
\[\frac{\sqrt{(x\ -\ x_1)^2\ +\ (y\ -\ y_1)^2}}{\pm\ \frac{a\ x\ +\ b\ y\ +\ c}{\sqrt{a^2\ +\ b^2}}}\ =\ 1\ \hspace{10cm}\]
\[\text{Squaring on both sides}\ \hspace{15cm}\]
\[[\sqrt{(x\ -\ x_1)^2\ +\ (y\ -\ y_1)^2}]^2\ =\ e^2\ [\frac{(ax\ +\ by\ +\ c)^2}{a^2\ +\ b^2}]\ \hspace{10cm}\]
\[\text{On simplification we get an equation of the second degree of the form}\]
\[ax^2\ +\ 2hxy\ +\ by^2\ +\ 2g\ x\ +\ 2f\ y +\ c\ =\ 0\]
Note:
\[\text{General equation of a conic}\ ax^2\ +\ 2hxy\ +\ by^2\ +\ 2g\ x\ +\ 2f\ y +\ c\ =\ 0\ represents\]
(i) a circle if a = b and h = 0.
(ii) a parabola if h2 = ab.
(iii) an ellipse if h2 < ab.
(iv) a hyperbola if h2 > ab.
\[\color {purple} {Example\ 1\ .}\ \color {red} {Prove\ that}\ the\ equation\ x^2\ +\ 6\ x\ y\ +\ 9y^2\ +\ 4\ x\ +\ 12\ y\ -\ 5\ =\ 0\ \hspace{10cm}\]\[\color {red} {is\ a\ parobala}\ \hspace{5cm}\]
\[\color {blue} {Soln:}\ x^2\ +\ 6\ x\ y\ +\ 9y^2\ +\ 4\ x\ +\ 12\ y\ -\ 5\ =\ 0\ ———- (1)\ \hspace{10cm}\]
\[\hspace{2cm}\ Condition\ for\ (1)\ to\ represent\ parabola\ is\ h^2\ =\ ab\ \hspace{8cm}\]
\[Comparing\ with\ a\ x^2\ +\ 2h\ x\ y\ +\ by^2\ +\ 2\ g\ x\ +\ 2\ f\ y\ +\ c\ = 0\]
\[We\ get\ a\ =\ 1,\ b\ =\ 9\ \hspace{5cm}\ 2h\ =\ 6,\ \implies\ h\ =\ 3\ \hspace{5cm}\]
\[h^2\ =\ ab\]
\[\therefore\ (1)\ represents\ a\ parabola\ \hspace{10cm}\]
VIDEO
\[\color {purple} {Example\ 2\ .}\ \color {red} {Show\ that}\ the\ equation\ x^2\ +\ 4y^2\ -\ 4\ x\ +\ 24\ y\ +\ 31\ =\ 0\ \hspace{10cm}\]\[\color {red} {represents\ an\ ellipse}\ \hspace{5cm}\]
\[\color {blue} {Soln:}\ x^2\ +\ 4y^2\ -\ 4\ x\ +\ 24\ y\ +\ 31\ =\ 0\ ———- (1)\ \hspace{10cm}\]
\[\hspace{2cm}\ Condition\ for\ (1)\ to\ represent\ an\ ellipse\ is\ h^2\ -\ ab\ \lt\ 0\ \hspace{8cm}\]
\[Comparing\ with\ a\ x^2\ +\ 2h\ x\ y\ +\ by^2\ +\ 2\ g\ x\ +\ 2\ f\ y\ +\ c\ = 0\]
\[We\ get\ a\ =\ 1,\ b\ =\ 4\ \hspace{5cm}\ 2h\ =\ 0,\ \implies\ h\ =\ 0\ \hspace{5cm}\]
\[\hspace{2cm}\ h^2\ -\ ab\ =\ (0)^2\ -\ 1(4)\ =\ -\ 4\ \lt\ 0\ \hspace{8cm}\]
\[\therefore\ (1)\ represents\ an\ ellipse\ \hspace{10cm}\]
VIDEO
\[\color {purple} {Example\ 3\ .}\ \color {red} {Check}\ whether\ the\ conic\ 2\ x^2\ -\ 16\ x\ y\ +\ 8\ y^2\ -\ y\ +\ 3\ = 0\ \color {red} {represent\ a\ hyperbola}\ \hspace{5cm}\]
\[\color {blue} {Soln:}\ 2\ x^2\ -\ 16\ x\ y\ +\ 8\ y^2\ -\ y\ +\ 3\ ———- (1)\ \hspace{10cm}\]
\[\hspace{2cm}\ Condition\ for\ (1)\ to\ represent\ hyperbola\ is\ h^2\ -\ ab\ \gt\ 0\ \hspace{8cm}\]
\[Comparing\ with\ a\ x^2\ +\ 2h\ x\ y\ +\ by^2\ +\ 2\ g\ x\ +\ 2\ f\ y\ +\ c\ = 0\]
\[We\ get\ a\ =\ 2,\ b\ =\ 8\ \hspace{5cm}\ 2h\ =\ – 16,\ \implies\ h\ =\ -\ 8\ \hspace{5cm}\]
\[\hspace{2cm}\ h^2\ -\ ab\ =\ (-8)^2\ -\ 2(8)\ =\ 64\ -\ 16\ =\ 48\ \gt\ 0\ \hspace{8cm}\]
\[\therefore\ (1)\ represents\ a\ hyperbola\ \hspace{10cm}\]
VIDEO
\[\color {purple} {Example\ 4\ .}\ \color {red} {Find\ the\ equation\ of\ the\ parabola}\ with\ focus\ at\ (1,\ -1)\ \hspace{10cm}\]\[and\ directrix\ x\ -\ y\ =\ 0.\ \hspace{5cm}\]
\[\color {blue} {Soln:}\ For\ parabola\ e\ =\ 1\ \hspace{15cm}\]
\[\hspace{2cm}\ Given\ Focus\ is\ S(1,\ -\ 1)\ and\ directrix\ is\ x\ -\ y\ =\ 0.\ \hspace{8cm}\]
\[Always\ \frac{SP}{PM}\ =\ e\ =\ 1\ \hspace{10cm}\]
\[\frac{\sqrt{(x\ -\ x_1)^2\ +\ (y\ -\ y_1)^2}}{\pm\ \frac{a\ x\ +\ b\ y\ +\ c}{\sqrt{a^2\ +\ b^2}}}\ =\ 1\ \hspace{10cm}\]
\[\frac{\sqrt{(x\ -\ 1)^2\ +\ (y\ +\ 1)^2}}{\pm\ \frac{x\ -\ y}{\sqrt{(1)^2\ +\ (-1)^2}}}\ =\ 1\ \hspace{10cm}\]
\[\sqrt{(x\ -\ 1)^2\ +\ (y\ +\ 1)^2}\ =\ \pm\ \frac{x\ -\ y}{\sqrt{2}}\ \hspace{10cm}\]
\[(x\ -\ 1)^2\ +\ (y\ +\ 1)^2\ =\ \frac{(x\ -\ y)^2}{2}\ \hspace{10cm}\]
\[\hspace{2cm}\ 2(x^2\ -\ 2\ x\ +\ 1\ +\ y^2\ +\ 2\ y\ +\ 1)\ =\ x^2\ +\ y^2\ -\ 2\ x\ y\ \hspace{8cm}\]
\[\hspace{2cm}\ 2\ x^2\ -\ 4\ x\ +\ 2\ +\ 2\ y^2\ +\ 4\ y\ +\ 2\ -\ x^2\ -\ y^2\ +\ 2\ x\ y\ =\ 0\ \hspace{10cm}\]
\[\hspace{2cm}\ 2\ x^2\ -\ 4\ x\ +\ 2\ +\ 2\ y^2\ +\ 4\ y\ +\ 2\ -\ x^2\ -\ y^2\ +\ 2\ x\ y\ =\ 0\ \hspace{8cm}\]
\[\hspace{2cm}\ \boxed {x^2\ +\ 2\ x\ y\ -\ 4\ x\ +\ y^2\ +\ 4\ y\ +\ 4\ =\ 0}\ \hspace{8cm}\]
VIDEO
\[\color {purple} {Example\ 5\ .}\ \color {red} {Find\ the\ equation\ of\ the\ parabola}\ with\ focus\ at\ (2,\ 1)\ \hspace{10cm}\]\[and\ directrix\ 2x\ +\ y\ +\ 1\ =\ 0.\ \hspace{5cm}\]
\[\color {blue} {Soln:}\ For\ parabola\ e\ =\ 1\ \hspace{15cm}\]
\[\hspace{2cm}\ Given\ Focus\ is\ S(2,\ 1)\ and\ directrix\ is\ 2\ x\ +\ y\ +\ 1\ =\ 0.\ \hspace{8cm}\]
\[Always\ \frac{SP}{PM}\ =\ e\ =\ 1\ \hspace{10cm}\]
\[\frac{\sqrt{(x\ -\ x_1)^2\ +\ (y\ -\ y_1)^2}}{\pm\ \frac{a\ x\ +\ b\ y\ +\ c}{\sqrt{a^2\ +\ b^2}}}\ =\ 1\ \hspace{10cm}\]
\[\frac{\sqrt{(x\ -\ 2)^2\ +\ (y\ -\ 1)^2}}{\pm\ \frac{2x\ +\ y\ +\ 1}{\sqrt{(2)^2\ +\ (1)^2}}}\ =\ 1\ \hspace{10cm}\]
\[\sqrt{(x\ -\ 2)^2\ +\ (y\ -\ 1)^2}\ =\ \pm\ \frac{2x\ +\ y\ +\ 1}{\sqrt{5}}\ \hspace{10cm}\]
\[(x\ -\ 2)^2\ +\ (y\ -\ 1)^2\ =\ \frac{(2x\ +\ y\ +\ 1)^2}{5}\ \hspace{10cm}\]
\[\hspace{2cm}\ 5(x^2\ -\ 4\ x\ +\ 4\ +\ y^2\ -\ 2\ y\ +\ 1)\ =\ 4x^2\ +\ y^2\ +\ 1\ +\ 4\ x\ y\ + 4\ x\ +\ 2y\ \hspace{8cm}\]
\[\hspace{2cm}\ 5\ x^2\ -\ 20\ x\ +\ 20\ +\ 5\ y^2\ -\ 10\ y\ +\ 5\ -\ 4\ x^2\ -\ y^2\ -\ 1\ -\ 4\ x\ y\ -\ 4x\ -\ 2y\ =\ 0\ \hspace{10cm}\]
\[\hspace{2cm}\ x^2\ +\ 4\ y^2\ -\ 4\ x\ y\ -\ 24\ x\ -\ 12\ y\ +\ 24\ =\ 0\ \hspace{8cm}\]
\[\hspace{2cm}\ \boxed {x^2\ +\ 4\ y^2\ -\ 4\ x\ y\ -\ 24\ x\ -\ 12\ y\ +\ 24\ =\ 0}\ \hspace{8cm}\]
VIDEO
Standard equation of a parabola with vertex at the origin \[\text{Standard equation of a parabola with vertex at the origin is}\ y^2\ =\ 4a\ x\ \hspace{5cm}\]
Focus: The fixed point used to draw the parabola is called the focus (F). Here focus is F(a, 0)
Directrix : The fixed line used to draw a parabola is called the directrix of the parabola. Here the equation of the directrix is x = – a
Axis : The axis of the parabola is the axis of symmetry. The curve \[y^2 = 4ax\] is symmetrical about x – axis and hence x-axis or y = 0 is the axis of the parabola \[y^2 = 4ax\] . Note that the axis of the parabola passes through the focus and perpendicular to the directrix.
Vertex : The point of intersection of the parabola with its axis is called its vertex. Here the vertex V is (0, 0)
Focal distance: The distance between a point on the parabola and its focus is called a focal distance.
Focal chord: A chord which passes through the focus of the parabola is called the focal chord of the parabola.
Latus rectum : It is a focal chord perpendicular to the axis of the parabola. Here, the equation of the lotus rectum is x = a. Length of the lotus rectum is 4a.
(i). Vertex O (0, 0)
(ii). Focus S (a, 0)
(iii). Equation of axis is equation x – axis is y = 0.
(iv). LL’ – Length of lotus rectum = 4a
(v). Equation of lotus rectum. x = a
(vi). AB – Directrix
(vii). Equation of the directrix x = – a.
Standard equation of a parabola with vertex at (h , k) \[\text{Standard equation of a parabola with vertex at (h, k) is}\ (y\ -\ k)^2\ =\ 4a (x – h)\ \hspace{5cm}\]
\[\color {purple} {Example\ 6\ .}\ \color {red} {Find\ the\ equation\ of\ the\ Ellipse}\ with\ focus\ (2,\ 3)\ and\ directrix\ x\ =\ 7\ and\ e\ =\ \frac{1}{2}\ \hspace{5cm}\]
\[\color {blue} {Soln:}\ \hspace{20cm}\]
\[\hspace{2cm}\ Given\ Focus\ is\ S(2,\ 3)\ and\ directrix\ is\ x\ -\ 7\ =\ 0,\ e\ =\ \frac{1}{2}\ \hspace{8cm}\]
\[\frac{\sqrt{(x\ -\ x_1)^2\ +\ (y\ -\ y_1)^2}}{\pm\ \frac{a\ x\ +\ b\ y\ +\ c}{\sqrt{a^2\ +\ b^2}}}\ =\ e\ \hspace{10cm}\]
\[\frac{\sqrt{(x\ -\ 2)^2\ +\ (y\ -\ 3)^2}}{\pm\ \frac{x\ -\ 7}{\sqrt{(1)^2\ +\ (0)^2}}}\ =\ \frac{1}{2}\ \hspace{10cm}\]
\[2\ \sqrt{(x\ -\ 2)^2\ +\ (y\ -\ 3)^2}\ =\ \pm\ \frac{x\ -\ 7}{\sqrt{1}}\ \hspace{10cm}\]
\[4\ (x\ -\ 2)^2\ +\ (y\ -\ 3)^2\ =\ \frac{(x\ -\ 7)^2}{1}\ \hspace{10cm}\]
\[\hspace{2cm}\ 4(x^2\ -\ 4\ x\ +\ 4\ +\ y^2\ -\ 6\ y\ +\ 9)\ =\ x^2\ -\ 14\ x\ +\ 49\ \hspace{8cm}\]
\[\hspace{2cm}\ 4\ x^2\ -\ 16\ x\ +\ 16\ -\ 24\ y\ +\ 36\ -\ x^2\ +\ 14\ x\ -\ 49\ =\ 0\ \hspace{10cm}\]
\[\hspace{2cm}\ 3\ x^2\ -\ 2\ x\ +\ 4\ y^2\ -\ 24\ y\ +\ 3\ =\ 0\ \hspace{8cm}\]
\[\hspace{2cm}\ 3\ x^2\ +\ 4\ y^2\ – 2\ x\ -\ 24\ y\ +\ 3\ =\ 0\ \hspace{8cm}\]
VIDEO
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