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Animated Solution for Mathematics - Straight Lines: If the sum of the slopes of the lines given by $x^{2} - 2 c x y - 7 y^{2} = 0$ is four times their product $c$ has the value

Q: Wait, why do we divide by $x^2$? Can't we just solve for $y$ using the quadratic formula?

You absolutely could! Solving for $y$ in terms of $x$ using the quadratic formula would give you two linear equations. However, that path is algebraically messy and prone to calculation errors. By dividing by $x^2$, we transform the geometry into a simple quadratic in $m = y/x$, which is the language of slopes. It is the elegant, professional way to handle homogeneous equations.

Q: Why is the sum of slopes $-\frac{2h}{b}$ and not $-\frac{b}{a}$ like in a normal quadratic?

Great catch! In a standard quadratic $Ax^2 + Bx + C = 0$, the sum of roots is $-B/A$. Here, our 'variable' is the slope $m$. When you divide $ax^2 + 2hxy + by^2 = 0$ by $x^2$, you get $b(y/x)^2 + 2h(y/x) + a = 0$. If you let $m = y/x$, this becomes $bm^2 + 2hm + a = 0$. Applying the sum of roots formula here gives $- (2h) / b$. Always check your coefficients!

Q: Does this only work if the lines pass through the origin?

Yes! A homogeneous equation of the second degree, $ax^2 + 2hxy + by^2 = 0$, represents two lines that *must* pass through the origin. If there were constant terms or linear terms like $gx$ or $fy$, the lines would be shifted away from the origin, and this specific slope-sum formula would no longer apply directly.

Q: Is there a geometric interpretation of $c$ here?

Think of $c$ as a parameter that controls the 'spread' or the angle between the two lines. By changing $c$, you are rotating the lines around the origin. The condition $m_1 + m_2 = 4(m_1 m_2)$ is essentially a constraint on the orientation of these lines relative to each other.

Select Answer:

Visualized Solution

The Homogeneous Equation

Given equation: $x^{2} - 2 c x y - 7 y^{2} = 0$
Represents a pair of straight lines passing through the origin $(0, 0)$ .
Let their slopes be $m_{1}$ and $m_{2}$ .

Standard Form Comparison

Standard Form: $a x^{2} + 2 h x y + b y^{2} = 0$
We need to compare our equation with this standard form.

Extracting Coefficients

Comparing $x^{2} - 2 c x y - 7 y^{2} = 0$ with $a x^{2} + 2 h x y + b y^{2} = 0$ :
$a = 1$
$2 h = - 2 c \Rightarrow h = - c$
$b = - 7$

Sum of Slopes Formula

Formula for the sum of slopes: $m_{1} + m_{2} = - \frac{2 h}{b}$

Calculating Sum of Slopes

Substitute $2 h = - 2 c$ and $b = - 7$ :
$m_{1} + m_{2} = - \frac{- 2 c}{- 7}$
$m_{1} + m_{2} = - \frac{2 c}{7}$

Product of Slopes Formula

Formula for the product of slopes: $m_{1} m_{2} = \frac{a}{b}$

Calculating Product of Slopes

Substitute $a = 1$ and $b = - 7$ :
$m_{1} m_{2} = \frac{1}{- 7}$
$m_{1} m_{2} = - \frac{1}{7}$

The Given Condition

According to the problem:
Sum of slopes $= 4 \times$ (Product of slopes)
$m_{1} + m_{2} = 4 (m_{1} m_{2})$

Substituting into the Condition

Substitute the calculated expressions:
$- \frac{2 c}{7} = 4 (- \frac{1}{7})$

Simplifying the Equation

Equation: $- \frac{2 c}{7} = - \frac{4}{7}$
Multiply both sides by $- 7$ :
$2 c = 4$

Solving for $c$

Divide both sides by $2$ :
$c = \frac{4}{2}$
$c = 2$

Final Conclusion

The value of $c$ is $2$ .
Key Takeaway: For $a x^{2} + 2 h x y + b y^{2} = 0$ , always use $m_{1} + m_{2} = - \frac{2 h}{b}$ and $m_{1} m_{2} = \frac{a}{b}$ .

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The Sigma Insight: Various Forms of Equations of a Line

Solution Diagram

Analyzing the Setup

The equation

x^{2} - 2 c x y - 7 y^{2} = 0

is a homogeneous equation of the second degree. In coordinate geometry, such an equation represents a pair of straight lines that intersect at the origin

(0, 0)

To analyze these lines, we define their slopes as

m_{1}

and

m_{2}

. We can extract these slopes by dividing the original equation by

x^{2}

, assuming $x eq 0$.

This transformation yields:

1 - 2 c (\frac{y}{x}) - 7 (\frac{y}{x})^{2} = 0

The Slope Transformation

By substituting

m = \frac{y}{x}

, the equation simplifies into a quadratic form in terms of

m

- 7 m^{2} - 2 c m + 1 = 0

Multiplying by

- 1

to standardize the expression, we obtain:

7 m^{2} + 2 c m - 1 = 0

The roots of this quadratic equation,

m_{1}

and

m_{2}

, represent the slopes of the two lines passing through the origin.

Applying Vieta’s Formulas

We utilize Vieta’s formulas to relate the coefficients of the quadratic equation to the sum and product of its roots. For a quadratic

A m^{2} + B m + C = 0

, the sum of the roots is

- \frac{B}{A}

and the product is

\frac{C}{A}

Comparing our equation

7 m^{2} + 2 c m - 1 = 0

to the standard form, we identify

A = 7

B = 2 c

, and

C = - 1

. This gives us:

m_{1} + m_{2} = - \frac{2 c}{7}

m_{1} m_{2} = - \frac{1}{7}

The Final Calculation

The problem provides the constraint that the sum of the slopes is four times their product, expressed as:

m_{1} + m_{2} = 4 (m_{1} m_{2})

Substituting our derived values into this condition, we get:

- \frac{2 c}{7} = 4 (- \frac{1}{7})

By multiplying both sides by

- 7

, the equation simplifies to:

2 c = 4

Solving for

c

, we find the final result:

$c = 2$

Common Traps & FAQs

Wait, why do we divide by

x^{2}

? Can't we just solve for

y

using the quadratic formula?

You absolutely could! Solving for

y

in terms of

x

using the quadratic formula would give you two linear equations. However, that path is algebraically messy and prone to calculation errors. By dividing by

x^{2}

, we transform the geometry into a simple quadratic in

m = y / x

, which is the language of slopes. It is the elegant, professional way to handle homogeneous equations.

Why is the sum of slopes

- \frac{2 h}{b}

and not

- \frac{b}{a}

like in a normal quadratic?

Great catch! In a standard quadratic

A x^{2} + B x + C = 0

, the sum of roots is

- B / A

. Here, our 'variable' is the slope

m

. When you divide

a x^{2} + 2 h x y + b y^{2} = 0

x^{2}

, you get

b (y / x)^{2} + 2 h (y / x) + a = 0

. If you let

m = y / x

, this becomes

b m^{2} + 2 hm + a = 0

. Applying the sum of roots formula here gives

- (2 h) / b

. Always check your coefficients!

Does this only work if the lines pass through the origin?

Yes! A homogeneous equation of the second degree,

a x^{2} + 2 h x y + b y^{2} = 0

, represents two lines that *must* pass through the origin. If there were constant terms or linear terms like

g x

f y

, the lines would be shifted away from the origin, and this specific slope-sum formula would no longer apply directly.

What if the lines are coincident? Does the formula still hold?

It does! If the lines are coincident, the discriminant of the quadratic in

m

will be zero, meaning

m_{1} = m_{2}

. The formulas for the sum and product of roots are universal properties of quadratic equations, regardless of whether the roots are distinct, real, or even complex.

Is there a geometric interpretation of

c

here?

Think of

c

as a parameter that controls the 'spread' or the angle between the two lines. By changing

c

, you are rotating the lines around the origin. The condition

m_{1} + m_{2} = 4 (m_{1} m_{2})

is essentially a constraint on the orientation of these lines relative to each other.

Animated Solution for Mathematics - Straight Lines: If the sum of the slopes of the lines given by $x^{2} - 2 c x y - 7 y^{2} = 0$ is four times their product $c$ has the value

Select Answer:

Visualized Solution

The Homogeneous Equation

Standard Form Comparison

Extracting Coefficients

Sum of Slopes Formula

Calculating Sum of Slopes

Product of Slopes Formula

Calculating Product of Slopes

The Given Condition

Substituting into the Condition

Simplifying the Equation

Solving for $c$

Final Conclusion

The Sigma Insight: Various Forms of Equations of a Line

Analyzing the Setup

The Slope Transformation

Applying Vieta’s Formulas

The Final Calculation

Similar Questions

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Animated Solution for Mathematics - Straight Lines: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the sum of the slopes of the lines given by x2−2cxy−7y2=0 is four times their product c has the value

Select Answer:

Visualized Solution

The Homogeneous Equation

Standard Form Comparison

Extracting Coefficients

Sum of Slopes Formula

Calculating Sum of Slopes

Product of Slopes Formula

Calculating Product of Slopes

The Given Condition

Substituting into the Condition

Simplifying the Equation

Solving for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } c

Final Conclusion

The Sigma Insight: Various Forms of Equations of a Line

Analyzing the Setup

The Slope Transformation

Applying Vieta’s Formulas

The Final Calculation

Similar Questions

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Animated Solution for Mathematics - Straight Lines: If the sum of the slopes of the lines given by $x^{2} - 2 c x y - 7 y^{2} = 0$ is four times their product $c$ has the value

Solving for $c$

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