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.
General equation of a conic ax2 + 2hxy + by2 + 2gx + 2fy + 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.
PART – A
\[Example\ 1:\ Prove\ that\ the\ equation\ x^2\ +\ 6\ x\ y\ +\ 9y^2\ +\ 4\ x\ +\ 12\ y\ -\ 5\ =\ 0\ \hspace{10cm}\]\[is\ a\ parobala\ \hspace{5cm}\]
Soln: x2 + 6xy + 9y2 + 4x + 12y – 5 = 0 —————– ( 1 )
Condition for ( 1 ) to represent parabola is h2 = ab
From ( 1 ) a = 1, b = 9
2h = 6 ⇒ h = 3
h2 = ab
32 = 1 ( 9)
9 = 9. ∴ ( 1 ) represents a parabola.
Example 2: Show that the equation x2 + 4y2 – 4x +24y + 31 = 0 represents an
ellipse.
Soln: Given x2 + 4y2 – 4x +24y + 31 = 0 –––– (1)
ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 ––––– (2)
Comparing, we get
a = 1 2h = 0 b = 4
h =0
h2 – ab = (0)2 – 1(4) = 0 – 4 = -4 < 0
Given equation ( 1 ) represents ellipse.
Example 3:
\[\color{Red}{Check\ whether\ the\ conic\ 2\ x^2\ -\ 16\ x\ y\ +\ 8\ y^2\ -\ y\ +\ 3\ = 0\ represent\ a\ hyperbola}\ \hspace{5cm}\]
\[Soln:\ 2\ x^2\ -\ 16\ x\ y\ +\ 8\ y^2\ -\ y\ +\ 3\ ———- (1)\ \hspace{10cm}\]
\[\hspace{2cm}\ Condition\ for\ (1)\ to\ represent\ hyperbola\ is\ h^2\ -\ ab\ \gt\ 0\ \hspace{8cm}\]
\[Comparing\ with\ a\ x^2\ +\ 2h\ x\ y\ +\ by^2\ +\ 2\ g\ x\ +\ 2\ f\ y\ +\ c\ = 0\]
\[We\ get\ a\ =\ 2,\ b\ =\ 8\ \hspace{5cm}\ 2h\ =\ – 16,\ \implies\ h\ =\ -\ 8\ \hspace{5cm}\]
\[\hspace{2cm}\ h^2\ -\ ab\ =\ (-8)^2\ -\ 2(4)\ =\ 64\ -\ 8\ =\ 56\ \gt\ 0\ \hspace{8cm}\]
\[\therefore\ (1)\ represents\ a\ hyperbola\ \hspace{10cm}\]
General Equation of a 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 aparabola.
(iii) If e>1, the conic is called a hyperbola.
Equation of a Conic with its focus at (x1 , y1 ) and the directrix ax + by + c = 0
\[Let\ the\ focus\ be\ S(x_1,\ y_1)\ and\ directrix\ be\ the\ line\ a\ x\ +\ b\ y\ +\ c\ =\ 0\ \hspace{10cm}\]
\[P(x,\ y)\ be\ any\ point\ on\ it\ \hspace{18cm}\]
\[SP\ =\ \sqrt{(x\ -\ x_1)^2\ +\ (y\ -\ y_1)^2}\ \hspace{18cm}\]
\[PM\ =\ Perpendicular\ distance\ of\ P\ from\ the\ line\ a\ x\ +\ b\ y\ +\ c\ =\ 0\ \hspace{10cm}\]
\[=\ \pm\ \frac{a\ x\ +\ b\ y\ +\ c}{\sqrt{a^2\ +\ b^2}}\ \hspace{15cm}\]
\[Example\ 1:\ Find\ the\ equation\ of\ the\ parabola\ with\ focus\ at\ (1,\ -1)\ \hspace{10cm}\]\[and\ directrix\ x\ -\ y\ =\ 0.\ \hspace{5cm}\]
\[Soln:\ For\ parabola\ e\ =\ 1\ \hspace{15cm}\]
\[\hspace{2cm}\ Given\ Focus\ is\ S(1,\ -\ 1)\ and\ directrix\ is\ x\ -\ y\ =\ 0.\ \hspace{8cm}\]
\[Always\ \frac{SP}{PM}\ =\ e\ =\ 1\ \hspace{10cm}\]
\[\frac{\sqrt{(x\ -\ x_1)^2\ +\ (y\ -\ y_1)^2}}{\pm\ \frac{a\ x\ +\ b\ y\ +\ c}{\sqrt{a^2\ +\ b^2}}}\ =\ 1\ \hspace{10cm}\]
\[\frac{\sqrt{(x\ -\ 1)^2\ +\ (y\ +\ 1)^2}}{\pm\ \frac{x\ -\ y}{\sqrt{(1)^2\ +\ (-1)^2}}}\ =\ 1\ \hspace{10cm}\]
\[\sqrt{(x\ -\ 1)^2\ +\ (y\ +\ 1)^2}\ =\ \pm\ \frac{x\ -\ y}{\sqrt{2}}\ \hspace{10cm}\]
\[(x\ -\ 1)^2\ +\ (y\ +\ 1)^2\ =\ \frac{(x\ -\ y)^2}{2}\ \hspace{10cm}\]
\[\hspace{2cm}\ 2(x^2\ -\ 2\ x\ +\ 1\ +\ y^2\ +\ 2\ y\ +\ 1)\ =\ x^2\ +\ y^2\ -\ 2\ x\ y\ \hspace{8cm}\]
\[\hspace{2cm}\ 2\ x^2\ -\ 4\ x\ +\ 2\ +\ 2\ y^2\ +\ 4\ y\ +\ 2\ -\ x^2\ -\ y^2\ +\ 2\ x\ y\ =\ 0\ \hspace{10cm}\]
\[\hspace{2cm}\ 2\ x^2\ -\ 4\ x\ +\ 2\ +\ 2\ y^2\ +\ 4\ y\ +\ 2\ -\ x^2\ -\ y^2\ +\ 2\ x\ y\ =\ 0\ \hspace{8cm}\]
\[\hspace{2cm}\ \boxed {x^2\ +\ 2\ x\ y\ -\ 4\ x\ +\ y^2\ +\ 4\ y\ +\ 4\ =\ 0}\ \hspace{8cm}\]
Example 2:
\[\color{red}{Find\ the\ equation\ of\ the\ Ellipse\ with\ focus\ (2,\ 3)\ and\ directrix\ x\ =\ 7\ and\ e\ =\ \frac{1}{2}}\ \hspace{5cm}\]
\[Soln:\ \hspace{20cm}\]
\[\hspace{2cm}\ Given\ Focus\ is\ S(2,\ 3)\ and\ directrix\ is\ x\ -\ 7\ =\ 0,\ e\ =\ \frac{1}{2}\ \hspace{8cm}\]
\[\frac{\sqrt{(x\ -\ x_1)^2\ +\ (y\ -\ y_1)^2}}{\pm\ \frac{a\ x\ +\ b\ y\ +\ c}{\sqrt{a^2\ +\ b^2}}}\ =\ e\ \hspace{10cm}\]
\[\frac{\sqrt{(x\ -\ 2)^2\ +\ (y\ -\ 3)^2}}{\pm\ \frac{x\ -\ 7}{\sqrt{(1)^2\ +\ (0)^2}}}\ =\ \frac{1}{2}\ \hspace{10cm}\]
\[2\ \sqrt{(x\ -\ 2)^2\ +\ (y\ -\ 3)^2}\ =\ \pm\ \frac{x\ -\ 7}{\sqrt{1}}\ \hspace{10cm}\]
\[4\ (x\ -\ 2)^2\ +\ (y\ -\ 3)^2\ =\ \frac{(x\ -\ 7)^2}{1}\ \hspace{10cm}\]
\[\hspace{2cm}\ 4(x^2\ -\ 4\ x\ +\ 4\ +\ y^2\ -\ 6\ y\ +\ 9)\ =\ x^2\ -\ 14\ x\ +\ 49\ \hspace{8cm}\]
\[\hspace{2cm}\ 4\ x^2\ -\ 16\ x\ +\ 16\ -\ 24\ y\ +\ 36\ -\ x^2\ +\ 14\ x\ -\ 49\ =\ 0\ \hspace{10cm}\]
\[\hspace{2cm}\ 3\ x^2\ -\ 2\ x\ +\ 4\ y^2\ -\ 24\ y\ +\ 3\ =\ 0\ \hspace{8cm}\]
\[\hspace{2cm}\ 3\ x^2\ +\ 4\ y^2\ – 2\ x\ -\ 24\ y\ +\ 3\ =\ 0\ \hspace{8cm}\]
\[Condition\ for\ General\ equation\ of\ a\ conic\ a\ x^2\ +\ 2\ h\ x\ y\ +\ b\ y^2\ +\ 2\ g\ x\ +\ 2\ f\ y\ +\ c\ =\ 0\ \hspace{7cm}\]\[to\ represent\ pair\ of\ straight\ lines\ is\ a\ b\ c\ +\ 2\ f\ g\ h\ -\ a\ f^2\ -\ b\ g^2\ -\ c\ h^2\ =\ 0\ \hspace{5cm}\]
\[Example\ 1:\ Prove\ that\ equation\ 6\ x^2\ +\ 13\ x\ y\ +\ 6\ y^2\ +\ 8\ x\ +\ 7\ y\ +\ 2\ =\ 0\ \hspace{7cm}\]\[represents\ a\ pair\ of\ straight\ lines\ \hspace{5cm}\]
\[Soln:\ Given\ 6\ x^2\ +\ 13\ x\ y\ +\ 6\ y^2\ +\ 8\ x\ +\ 7\ y\ +\ 2\ =\ 0\ ———-(1)\ \hspace{8cm}\]
\[\hspace{2cm}\ a\ x^2\ +\ 2\ h\ x\ y\ +\ b\ y^2\ +\ 2\ g\ x\ +\ 2\ f\ y\ +\ c\ =\ 0\ \hspace{10cm}\]
\[\hspace{2cm}\ comparing\ we\ get\ \hspace{8cm}\]
\[\hspace{2cm}\ a\ =\ 6\ \hspace{2cm}\ 2\ h\ =\ 13\ \hspace{2cm}\ b\ =\ 6\ \hspace{2cm}\ 2\ g\ =\ 8\ \hspace{2cm}\ 2\ f\ =\ 7\ \hspace{2cm}\ c\ =\ 2\]
\[\hspace{6cm}\ h\ =\ \frac{13}{2}\ \hspace{4cm}\ g\ =\ 4\ \hspace{4cm}\ f\ =\ \frac{7}{2}\ \hspace{4cm}\]
\[\hspace{2cm}\ To\ claim\ equation\ (1)\ represents\ a\ pair\ of\ straight\ lines\ \hspace{8cm}\]
\[\hspace{2cm}\ \therefore\ i.e\ a\ b\ c\ +\ 2\ f\ g\ h\ -\ a\ f^2\ -\ b\ g^2\ -\ c\ h^2\ =\ 0\ \hspace{8cm}\]
\[\hspace{2cm}\ L.H.S\ =\ a\ b\ c\ +\ 2\ f\ g\ h\ -\ a\ f^2\ -\ b\ g^2\ -\ c\ h^2\ \hspace{8cm}\]
\[\hspace{2cm}\ =\ 6\ (6)\ (2)\ +\ 2\ (\frac{7}{2})\ (4)\ (\frac{13}{2})\ -\ 6\ (\frac{7}{2}) ^2\ -\ 6(4)^2\ -\ 2\ (\frac{13}{2}) ^2\ \hspace{8cm}\]
\[\hspace{2cm}\ =\ 72\ +\ \frac{364}{2}\ -\ 6\ (\frac{49}{4})\ -\ 6\ (16)\ -\ 2\ (\frac{169}{4})\ \hspace{10cm}\]
\[ =\ 72\ +\ 182\ -\ \frac{294}{4}\ -\ 96\ -\ \frac{338}{4}\ \hspace{10cm}\]
\[ =\ 158\ -\ (\frac{294\ +\ 338}{4})\ \hspace{10cm}\]
\[ =\ 158\ -\ \frac{632}{4}\ \hspace{10cm}\]
\[ =\ 158\ -\ 158\ \hspace{10cm}\]
\[ =\ 0\ =\ R.\ H.\ S\ \hspace{10cm}\]
\[\hspace{2cm}\ \therefore\ the\ given\ equation\ (1)\ represents\ a\ pair\ of\ straighlt\ lines\ \hspace{8cm}\]
Example 2:
\[\color{red}{Show\ that\ the\ second\ degree\ equation}\ \hspace{5cm}\]\[\color{red}{12\ x^2\ +\ 7\ x\ y\ -\ 10\ y^2\ +\ 13\ x\ +\ 45\ y\ -\ 35\ =\ 0}\ \hspace{7cm}\]\[\color{red}{in\ x\ and\ y\ represents\ a\ pair\ of\ straight\ lines}\ \hspace{5cm}\]
\[Soln:\ Given\ 12\ x^2\ +\ 7\ x\ y\ -\ 10\ y^2\ +\ 13\ x\ +\ 45\ y\ -\ 35\ =\ 0\ ———-(1)\ \hspace{8cm}\]
\[\hspace{2cm}\ a\ x^2\ +\ 2\ h\ x\ y\ +\ b\ y^2\ +\ 2\ g\ x\ +\ 2\ f\ y\ +\ c\ =\ 0\ \hspace{8cm}\]
\[\hspace{2cm}\ comparing\ we\ get\ \hspace{8cm}\]
\[\hspace{2cm}\ a\ =\ 12\ \hspace{2cm}\ 2\ h\ =\ 7\ \hspace{2cm}\ b\ =\ -\ 10\ \hspace{2cm}\ 2\ g\ =\ 13\ \hspace{2cm}\ 2\ f\ =\ \ 45\ \hspace{2cm}\ c\ =\ -\ 35\]
\[\hspace{6cm}\ h\ =\ \frac{7}{2}\ \hspace{4cm}\ g\ =\ \frac{13}{2}\ \hspace{4cm}\ f\ =\ \frac{45}{2}\ \hspace{4cm}\]
\[\hspace{2cm}\ To\ claim\ equation\ (1)\ represents\ a\ pair\ of\ straight\ lines\ \hspace{8cm}\]
\[\hspace{2cm}\ \therefore\ i.e\ a\ b\ c\ +\ 2\ f\ g\ h\ -\ a\ f^2\ -\ b\ g^2\ -\ c\ h^2\ =\ 0\ \hspace{8cm}\]
\[\hspace{2cm}\ L.H.S\ =\ a\ b\ c\ +\ 2\ f\ g\ h\ -\ a\ f^2\ -\ b\ g^2\ -\ c\ h^2\ \hspace{8cm}\]
\[\hspace{2cm}\ =\ 12\ (-10)\ (-35)\ +\ 2\ (\frac{45}{2})\ (\frac{13}{2})\ (\frac{7}{2})\ -\ 12\ (\frac{45}{2}) ^2\ +\ 10\ (\frac{13}{2}) ^2\ +\ 35\ (\frac{7}{2}) ^2\ \hspace{8cm}\]
\[\hspace{2cm}\ =\ 4200\ +\ \frac{4095}{4}\ -\ 12\ (\frac{2025}{4})\ +\ 10\ (\frac{169}{4})\ +\ 35\ (\frac{49}{4})\ \hspace{10cm}\]
\[\hspace{2cm}\ =\ 4200\ +\ \frac{4095}{4}\ -\ 3(2025)\ +\ 5(\frac{169}{2})\ +\ \frac{1715}{4}\ \hspace{10cm}\]
\[\hspace{2cm}\ =\ 4200\ +\ \frac{4095}{4}\ -\ 6075\ +\ \frac{845}{2}\ +\ \frac{1715}{4}\ \hspace{10cm}\]
\[ =\ -\ 1875\ +\ (\frac{4095\ +\ 1690\ +\ 1715}{4})\ \hspace{10cm}\]
\[ =\ -\ 1875\ +\ (\frac{7500}{4})\ \hspace{10cm}\]
\[ =\ -\ 1875\ +\ 1875\ \hspace{10cm}\]
\[ =\ 0\ =\ R.\ H.\ S\ \hspace{10cm}\]
\[\hspace{2cm}\ \therefore\ the\ given\ equation\ (1)\ represents\ a\ pair\ of\ straighlt\ lines\ \hspace{8cm}\]
\[Example\ 3:\ Find\ c\ if\ \ 2\ x^2\ +\ 3\ x\ y\ -\ 2\ y^2\ -\ 5\ x\ +\ 5\ y\ +\ c\ =\ 0\ \hspace{7cm}\]\[represents\ a\ pair\ of\ straight\ lines\ \hspace{5cm}\]
\[Soln:\ Given\ 2\ x^2\ +\ 3\ x\ y\ -\ 2\ y^2\ -\ 5\ x\ +\ 5\ y\ +\ c\ =\ 0\ ———-(1)\ \hspace{8cm}\]
\[\hspace{2cm}\ a\ x^2\ +\ 2\ h\ x\ y\ +\ b\ y^2\ +\ 2\ g\ x\ +\ 2\ f\ y\ +\ c\ =\ 0\ \hspace{8cm}\]
\[comparing\ we\ get\ \hspace{10cm}\]
\[\hspace{2cm}\ a\ =\ 2\ \hspace{2cm}\ 2\ h\ =\ 3\ \hspace{2cm}\ b\ =\ -\ 2\ \hspace{2cm}\ 2\ g\ =\ -\ 5\ \hspace{2cm}\ 2\ f\ =\ 5\ \hspace{2cm}\ c\ =\ c\]
\[ \hspace{6cm}\ h\ =\ \frac{3}{2}\ \hspace{6cm}\ g\ =\ -\ \frac{5}{2}\ \hspace{4cm}\ f\ =\ \frac{5}{2}\ \hspace{4cm}\]
\[\hspace{2cm}\ Given\ that\ \ (1)\ represents\ a\ pair\ of\ straight\ lines\ \hspace{8cm}\]
\[\hspace{2cm}\ i.e\ a\ b\ c\ +\ 2\ f\ g\ h\ -\ a\ f^2\ -\ b\ g^2\ -\ c\ h^2\ =\ 0\ \hspace{8cm}\]
\[\hspace{2cm}\ 2\ (-2)\ (c)\ +\ 2\ (\frac{5}{2})\ (-\frac{5}{2})\ (\frac{3}{2})\ -\ 2\ (\frac{5}{2}) ^2\ +\ 2(-\frac{5}{2})^2\ -\ c\ (\frac{3}{2}) ^2\ =\ 0\ \hspace{10cm}\]
\[\hspace{2cm}\ -\ 4\ c\ -\ \frac{75}{4}\ -\ 2\ (\frac{25}{4})\ +\ 2\ (\frac{25}{4})\ -\ c\ (\frac{9}{4})\ =\ 0\ \hspace{10cm}\]
\[\hspace{2cm}\ \frac{-\ 16\ c\ -\ 75\ -\ 50\ +\ 50\ -\ 9\ c}{4}\ =\ 0\ \hspace{10cm}\]
\[-\ 25\ c\ -\ 75\ =\ 0\ \hspace{10cm}\]
\[25\ c\ =\ -\ 75\ \hspace{10cm}\]
\[\boxed {c\ =\ -\ 3}\ \hspace{10cm}\]
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