Question

If the slope of one of the lines represented by the equation $a{{x}^{2}}+2hxy+b{{y}^{2}}=0$ be $\lambda$ times that of the other, then
A.$4\lambda h=ab(1+\lambda )$
B.$\lambda h+ab{{(1+\lambda )}^{2}}$
C.$4\lambda {{h}^{2}}=ab{{(1+\lambda )}^{2}}$
D.None of these

Answer 1

Hint : The slope of a line is the steepness and direction of a non-vertical line. When a line rises from left to right, the slope is positive. When a line falls from left to right, the slope is negative. If $m$ represents the slope of a line and coordinates $(x_1, y_1)$ and $x_1$ = $x_2$ respectively, then the slope of the line is given by the following formula.
$m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$
lf $(x_1$ = $X_2)$, then the line is vertical and the slope is undefined. Every line has an equation that can be written in the standard form $Ax\text{ }+\text{ }By\text{ }=\text{ }C$ where$A$, $B$, and $C$ are three integers, and $A$ and B are not both zero. A must be positive.

Complete step-by-step answer :
Given that,
$a{{x}^{2}}+2hxy+b{{y}^{2}}=0$
Let us assume that $m$ is the slope.
Then according to the question other slope will be given as $\lambda m$
As we know that the sum of the slope is given as
${{m}_{1}}~+\text{ }{{m}_{2}}~=\dfrac{-2h}{b}$
Using the above equation we get
$m+\text{ }\lambda m~=\dfrac{-2h}{b}$
Taking $m$ as common factor from LHS we get
$m(1+\text{ }\lambda )=\dfrac{-2h}{b}$
Transposing we get
$m=\dfrac{-2h}{b(1+\text{ }\lambda )}$
As we know the product of the slope is given as
$~{{m}_{1}}{{m}_{2}}~=\dfrac{a}{b}$
Further equating we get
$\lambda {{m}^{2}}~=\dfrac{a}{b}$
Simplifying the equation we get
${{m}^{2}}~=\dfrac{a}{b\lambda }$
Substituting the value of $m$ in the above equation we get
$\dfrac{4{{h}^{2}}}{{{b}^{2}}{{(1+\lambda )}^{2}}}=\dfrac{a}{b\lambda }$
Rearranging the equation we get
$4\lambda {{h}^{2}}=ab{{(1+\lambda )}^{2}}$
Therefore, option $C$ is the correct answer.
So, the correct answer is “Option C”.

Note : The order in which the points are taken really doesn't matter, as long as you subtract the x-values in the same order as you subtracted the y-values. A line is a curve in which every point on the line segment joining any two points on it lies on it.