Pair of straight lines — quick review for IITJEE Math

Pair of Straight Lines:

I: Equation of Family of Lines:

A first degree equation ax+by+c=0 represents a straight line involving three constants a, b and c, which can be reduced to two by dividing both the sides of the equation by a non-zero constant. For instance, if a \neq 0 we can write the equation as

x+\frac{b}{a}y+\frac{c}{a}=0 or x+By+C=0

So, in order to determine an equation of a line, we need two conditions on the line to determine these constants. For instance, if we know two points on the line or a point on the line and its slope etc., we know the line. But, if we know just one condition, we have infinite number of lines satisfying the given condition. In this case, the equation of the line contains an arbitrary constant and for different values of the constant we have different lines satisfying the given condition and the constant is called a parameter.

Some Equations of Family of Lines:

  1. Family of lines with given slope. (Family of given lines) y = mx +k, where k is a parameter represents a family of lines in which each line has slope m.
  2. Family of lines through a point: y-y_{0}=k(x-x_{0}), where k is a parameter represents a family of lines in which each line passes through the point (x_{0},y_{0}).
  3. Family of lines parallel to a given line: ax + by + k=0, where k is a parameter represents a family of lines which are parallel to the line ax+by+c=0.
  4. Family of lines through intersection of two given lines:
    a_{1}x+b_{1}y+c_{1}+k(a_{2}x+b_{2}y+c_{2})=0, where k is a parameter represents a family of lines, in which each line passes through the point of intersection of two intersecting llines a_{1}x+b_{1}y+c_{1}=0 and a_{2}x+b_{2}y+c_{2}=0.
  5. Family of lines perpendicular to a given line:
    bx-ay+c=0 where k is a parameter represents a family of lines, in which each line is perpendicular to the line ax+by+c=0.
  6. Family of lines making a given intercept on axes:
    (a) \frac{x}{a}+\frac{y}{k}=1, where k is a parameter, represents a family of lines, in which each line makes an intercept a on the axis of x. and,

6(b) \frac{x}{k} + \frac{y}{b}=1, where k is a parameter, represents a family of lines, in which each line makes an intercept b on the axis of y.

7. Family of lines at a constant distance from the origin:

x\cos {\alpha}+y\sin{\alpha}=p, where \alpha is a parameter, represents a family of lines, in which each line is at a distance p from the origin.

II: Pair of Straight Lines:

A) Pair of straight lines through the origin: A homogeneous equation of the second degree ax^{2}+2hxy+by^{2}=0…call this equation 1; represents a pair of straight lines passing through the origin if and only if h^{2}-ab \geq 0. If b \neq 0, the given equation can be written as

y^{2}+\frac{2h}{b}xy+\frac{a}{b}x^{2}=0, or (\frac{y}{x})^{2}+\frac{2h}{b}(\frac{y}{x})+\frac{a}{b}=0, which, being quadratic in y/x, gives two values of y/x, say m_{1} and m_{2} and hence, the equations y=m_{1}x and y=m_{2}x of two straight lines passing through the origin. The slopes m_{1} and m_{2} of these straight lines are given by the relations

m_{1}+m_{2}=-\frac{2h}{b}…call this equation (2).

m_{1}m_{2}=\frac{a}{b}…call this equation (3).

B) Angle between the lines: represented by ax^{2}+2hxy+by^{2}=0. If \theta is an angle, between the lines represented by (1), then

\tan{\theta} = \frac{m_{1}-m_{2}}{1+m_{1}m_{2}}=\frac{\pm\sqrt{(m_{1}+m_{2})^{2}-4m_{1}m_{2}}}{1+m_{1}m_{2}} = \frac{\pm \sqrt{4h^{2}-4ab}}{a+b}

\Longrightarrow \theta = \arctan (\frac{\pm 2\sqrt{h^{2}-ab}}{a+b})

If (a+b)=0, the lines are perpendicular, and if h^{2}=ab, then the lines are coincident.

C) Equation of the bisectors of the angles between the lines ax^{2}+2hxy=by^{2}=0.

Let y=m_{1}x and y=m_{2}x be the equation of the straight lines represented by the given equation. Then, equations of the angle bisectors are:

\frac{y-m_{1}x}{\sqrt{1+m_{1}^{2}}} = \frac{y-m_{2}x}{\sqrt{1+m_{1}^{2}}}

The joint equation of  these bisectors can be written as

(\frac{y-m_{1}x}{\sqrt{1+m_{1}^{2}}} + \frac{y-m_{2}x}{1+m_{2}^{2}})(\frac{y-m_{1}x}{1+m_{1}^{2}} - \frac{y-m_{2}x}{1+m_{2}^{2}})=0

or, (1+m_{2}^{2})(y-m_{1}x)^{2} - (1+m_{1}^{2})(y-m_{2}x)^{2}=0

or, x^{2}(m_{1}^{2}-m_{2}^{2}) - 2xy(m_{1}-m_{2})(1-m_{1}m_{2}) + y^{2}(m_{2}^{2}-m_{1}^{2})=0

or, \frac{x^{2}-y^{2}}{2xy} = \frac{1-m_{1}m_{2}}{m_{1}+m_{2}} = \frac{a-b}{2h}….this comes from (2) and (3).

or, \frac{x^{2}-y^{2}}{a-b} = \frac{xy}{h}

D) The general equation of second degree:

ax^{2}+2hxy+by^{2}+2gx+2fy+c=0 represents a pair of straight lines if and only if

abc + 2fgh - af^{2} -bg^{2} - ch^{2} =0, that is, if

\left | \begin{array}{ccc}    a & h & g \\    h & b & f \\    g & f & c \end{array} \right |=0

If h^{2}=ab, the co-ordinates (x_{1},y_{1}) of the point of intersection of these lines are obtained by solving the equations

ax_{1}+hy_{1}+x=0 and hx_{1}+by_{1}+f=0

\Longrightarrow x_{1}=\frac{hf-bg}{ab-h^{2}} and y_{1}=\frac{gh-af}{ab-h^{2}}

E) Equation ax^{2}+2hxy +by^{2}=0 represents a pair of straight lines through the origin parallel to the lines given by the general equation of second degree given in (4) and hence, the angle between the lines given in (4) is same as in (2), that is,

\theta = \arctan {\frac{\pm {h^{2}-ab}}{a+b}}

so that if a+b=0, the lines are perpendicular and if h^{2}=ab, the lines are parallel.

F) Equation of the lines joining the origin to the points of intersection of a line and a conic:

Let L \equiv lx + my + n=0

and S \equiv ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c=0

be the equations of a line and a conic, respectively. Writing the equation of the line as

\frac{lx + my}{-n} = 1 and making S=0 homogeneous with its help, we get S=ax^{2}+2hxy+by^{2}+2(gx +fy)

(\frac{lx+my}{-n}) + c(\frac{lx+my}{-n})^{2}=0, which being a homogeneous equation of second degree, represents a pair of straight lines through the origin and passing through the points common to S=0 and L=0.

G) Some important results regarding pair of lines:

(i) Equation of the pair of lines through the origin perpendicular to the pair of lines ax^{2}+2hxy+by^{2}=0 is bx^{2}+2hxy +ay^{2}=0.

(ii) The product of the perpendicular lines drawn from the point (x_{1}, y_{1}) on the lines ax^{2}+2hxy+by^{2}=0 is |\frac{ax_{1}^{2}+2hx_{1}y_{1}+by_{1}^{2}}{\sqrt{(a-b)^{2}+4h^{2}}}|

(iii) The product of the perpendiculars drawn from the origin to the lines ax^{2}+2hxy + by^{2}+2gx+2fy+c=0 is \frac{c}{\sqrt{(a-b)^{2}+4h^{2}}}

(iv) Equations of the bisectors of the angles between the lines represented by ax^{2}+2hxy + by^{2} +2gx + 2fy + c=0 are given by

\frac{(x-x_{1})^{2}-(y-y_{1})^{2}}{a-b} = \frac{(x-x_{1})(y-y_{1})}{h} where (x_{1}, y_{1}) is the point of intersection of the lines represented by the given equation.

(v) If ax^{2}+2hxy + by^{2}+2gx + 2fy + c=0 represents two parallel straight lines, then the distance between them is 2\sqrt{\frac{g^{2}-ac}{a(a+b)}}

Every nth degree homogeneous equation a_{0}x^{n}+a_{1}x^{n-1}y + a_{2}x^{n-2}y + \ldots + a_{n-1}xy^{n-1}+a_{n}y^{n}=0, then

m_{1}+m_{2}+ \ldots + m_{n} = - \frac{a_{n-1}}{a_{n}}

and m_{1}m_{2}\ldots m_{n}=(-1)^{n}\frac{a_{0}}{a_{n}}

To be continued later,

-Nalin Pithwa.

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