Question

If the slope of one of the lines represented by \[a{x^2} + 2hxy + b{y^2} = 0\] is the square of the other, then \[\dfrac{{(a + b)}}{h} + \dfrac{{8{h^2}}}{{ab}}\] is
A). \[3\]
B). \[4\]
C). \[5\]
D). \[6\]

Answer 1

Hint: Here we are asked to find the value of the given expression from the given slope of one line. We will use the general equation of slope of a line that passes through the origin to find the sum and product of the slopes. Then we will try to get the given expression from those to find its value.
Formula Used: If the pair of lines is \[a{x^2} + 2hxy + b{y^2} = 0\] then
the sum of its slope\[ = \dfrac{{ - 2h}}{b}\] and the product of its slope\[ = \dfrac{a}{b}\].
\[{\left( {a + b} \right)^3} = {a^3} + {b^3} + 3ab\left( {a + b} \right)\]

Complete step-by-step solution:
It is given that one of the lines is represented by \[a{x^2} + 2hxy + b{y^2} = 0\], let us mark this equation as one.
\[a{x^2} + 2hxy + b{y^2} = 0........(1)\]
We know that the general form of a slope of a line is written as \[m = \dfrac{y}{x}\] Let us mark this as two.
\[m = \dfrac{y}{x}........(2)\]
Considering equation \[(1)\] and dividing it by \[{x^2}\] we get
\[\dfrac{{a{x^2}}}{{{x^2}}} + \dfrac{{2hxy}}{{{x^2}}} + \dfrac{{b{y^2}}}{{{x^2}}} = 0\]
Which can be re-written as
\[a + 2h\left( {\dfrac{y}{x}} \right) + b{\left( {\dfrac{y}{x}} \right)^2} = 0\]
Substituting equation \[(2)\] in the above equation we get
\[a + 2hm + b{m^2} = 0\]
Let \[{m_1}\] be the slope of one line and \[{m_2}\] be the slope of the other line. Since it is given that the slope of one line is a square of another, we can write it as
\[ {m_1} = m \\
  {m_2} = {m^2} \]
We know that the sum of the slope \[ = \dfrac{{ - 2h}}{b}\]. Thus, \[{m_1} + {m_2} = m + {m^2} = \dfrac{{ - 2h}}{b}\]
Also, we know that the product of the slope \[ = \dfrac{a}{b}\]. Thus, \[{m_1}{m_2} = m.{m^2} = {m^3} = \dfrac{a}{b}\]
Now consider \[{\left( {m + {m^2}} \right)^3}\]
\[{\left( {m + {m^2}} \right)^3} = {\left( {\dfrac{{ - 2h}}{b}} \right)^3}\]
Let us expand the above using the formula, \[{\left( {a + b} \right)^3} = {a^3} + {b^3} + 3ab\left( {a + b} \right)\] we get
\[{m^3} + {m^6} + 3m{m^2}\left( {m + {m^2}} \right) = {\left( {\dfrac{{ - 2h}}{b}} \right)^3}\]
On simplifying this we get
\[{m^3} + {m^6} + 3m{m^2}\left( {m + {m^2}} \right) = \dfrac{{ - 8{h^3}}}{{{b^3}}}\]
We know that \[{m^3} = \dfrac{a}{b}\] substituting it in the above we get
\[\dfrac{a}{b} + \dfrac{{{a^2}}}{{{b^2}}} + 3\left( {\dfrac{{ - 2h}}{b}} \right)\left( {\dfrac{a}{b}} \right) = \dfrac{{ - 8{h^3}}}{{{b^3}}}\]
On simplifying this further we get
\[\dfrac{a}{{{b^2}}}\left( {a + b} \right) + \dfrac{{8{h^3}}}{{{b^3}}} = \dfrac{{6ah}}{{{b^2}}}\]
Multiplying the above by \[{b^2}\] we get
\[a\left( {a + b} \right) + \dfrac{{8{h^3}}}{b} = 6ah\]
Now dividing the above by \[ah\] we get
\[\dfrac{{\left( {a + b} \right)}}{h} + \dfrac{{8{h^2}}}{{ab}} = 6\]
Thus, we got the value of the expression \[\dfrac{{(a + b)}}{h} + \dfrac{{8{h^2}}}{{ab}}\] as \[6\]. Now let us see the options.
Option (1) \[3\] is not the correct option as we got the value of the given expression is \[6\].
Option (2) \[4\] is not the correct option as we got the value of the given expression is \[6\].
Option (3) \[5\] is not the correct option as we got the value of the given expression is \[6\].
Option (4) \[6\] is the correct answer as we got the same value in our calculation.
Hence, option (4) \[6\] is the correct option.

Note: Here we are asked to find the value of a given expression so we have used the slopes of the given two lines to form the given expression to find its value. Here we have used the sum of the slopes as we know the value of that which made our calculation easier.