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Animated Solution for Mathematics - Conic Sections: Let the point $P$ of the focal chord $P Q$ of the parabola $y^{2} = 16 x$ be $(1, - 4)$ . If the focus of the parabola divides the chord $P Q$ in the ratio $m : n$ , $gcd (m, n) = 1$ , then $m^{2} + n^{2}$ is equal to :

Q: Can I use the slope formula instead of the section formula to find the ratio?

Absolutely! Since the focus $S$, point $P$, and point $Q$ are collinear, the slope of $SP$ must equal the slope of $SQ$. You can set $\frac{y_S - y_P}{x_S - x_P} = \frac{y_Q - y_S}{x_Q - x_S}$. This will give you the same ratio $m:n$. Both methods are valid, but the section formula is often more direct when you are already working with coordinates.

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Visualized Solution

Equation of the Parabola

Given Parabola: $y^{2} = 16 x$
Standard Form: $y^{2} = 4 a x$
Comparing coefficients: $4 a = 16 \Rightarrow a = 4$

Locating the Focus $S$

Focus of $y^{2} = 4 a x$ is $S (a, 0)$
Substituting $a = 4$ : $S (4, 0)$

Parametric Coordinates of $P$

Point $P (1, - 4)$ lies on the parabola.
Parametric form of any point: $(a t^{2}, 2 a t)$
For our parabola: $P (4 t^{2}, 8 t)$

Finding Parameter $t_{P}$

Equating y-coordinates: $8 t_{P} = - 4$
Solving for $t_{P}$ : $t_{P} = - \frac{1}{2}$
Verification with x-coordinate: $4 (- \frac{1}{2})^{2} = 1$

Property of the Focal Chord

A focal chord passes through the focus $S$ .
Let the other end be $Q (t_{Q})$ .
Key Property: $t_{P} \cdot t_{Q} = - 1$

Finding Parameter $t_{Q}$

Substitute $t_{P} = - \frac{1}{2}$ into the relation.
$(- \frac{1}{2}) \cdot t_{Q} = - 1$
Solving for $t_{Q}$ : $t_{Q} = 2$

Coordinates of Point $Q$

Parametric coordinates of $Q$ : $(4 t_{Q}^{2}, 8 t_{Q})$
Substitute $t_{Q} = 2$ :
$Q = (4 (2)^{2}, 8 (2))$

Calculating Coordinates of $Q$

$x_{Q} = 4 (4) = 16$
$y_{Q} = 8 (2) = 16$
Point $Q$ is $(16, 16)$

Applying the Section Formula

The focus $S (4, 0)$ divides $P Q$ in the ratio $m : n$ .
Section Formula for x-coordinate:
$x_{S} = \frac{m \cdot x _{Q} + n \cdot x _{P}}{m + n}$

Substituting Values into Section Formula

$x_{S} = 4$ , $x_{P} = 1$ , $x_{Q} = 16$
Substitute into the formula:
$4 = \frac{m ( 16 ) + n ( 1 )}{m + n}$

Solving for the Ratio $m : n$

Cross-multiply: $4 (m + n) = 16 m + n$
Expand: $4 m + 4 n = 16 m + n$
Rearrange: $3 n = 12 m$
Ratio: $\frac{m}{n} = \frac{3}{12} = \frac{1}{4}$

Final Calculation of $m^{2} + n^{2}$

We have $\frac{m}{n} = \frac{1}{4}$
Given $gcd (m, n) = 1$ , so $m = 1$ and $n = 4$ .
Calculate: $m^{2} + n^{2} = 1^{2} + 4^{2}$
$1 + 16 = 17$

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The Sigma Insight: Standard Equations of Parabola, Ellipse, and Hyperbola

Solution Diagram

Analyzing the Setup

Welcome, fellow traveler of the coordinate plane. Today, we are uncovering the hidden symmetries of the parabola

y^{2} = 16 x

By comparing

y^{2} = 16 x

to the standard form

y^{2} = 4 a x

, we identify

4 a = 16

, which gives us

a = 4

. This parameter

a

is the heartbeat of our parabola, placing the focus

S

(4, 0)

The Parametric Key

Consider the point

P (1, - 4)

. Any point on this parabola can be elegantly described using a single parameter

t

(a t^{2}, 2 a t)

With

a = 4

, our point

P

(4 t_{P}^{2}, 8 t_{P})

. By equating the

y

-coordinates,

8 t_{P} = - 4

, we find:

t_{P} = - \frac{1}{2}

The Focal Chord Property

We are dealing with a focal chord

P Q

, a line segment that passes through the focus

S

. There is a profound property for any focal chord of a parabola: the product of the parameters of its endpoints is always

- 1

That is,

t_{P} \cdot t_{Q} = - 1

. Since we know

t_{P} = - \frac{1}{2}

, we can instantly find

t_{Q}

(- \frac{1}{2}) \cdot t_{Q} = - 1 \Rightarrow t_{Q} = 2

This relationship is a bridge that connects the two ends of the chord without needing to calculate the equation of the line itself.

Finding the Other End

With

t_{Q} = 2

, we find the coordinates of

Q

using the parametric form

(4 t^{2}, 8 t)

. Substituting

t_{Q} = 2

x_{Q} = 4 (2)^{2} = 16

y_{Q} = 8 (2) = 16

Thus, our point

Q

(16, 16)

The Final Ratio

The focus

S (4, 0)

divides the chord

P Q

in the ratio

m : n

. We use the section formula for the

x

-coordinate:

x_{S} = \frac{m \cdot x _{Q} + n \cdot x _{P}}{m + n}

Substituting our values:

4 = \frac{m ( 16 ) + n ( 1 )}{m + n}

Cross-multiplying gives

4 m + 4 n = 16 m + n

, which simplifies to

3 n = 12 m

, or:

\frac{m}{n} = \frac{3}{12} = \frac{1}{4}

Since

gcd (m, n) = 1

, we have

m = 1

and

n = 4

. The final calculation is:

$m^{2} + n^{2} = 1^{2} + 4^{2} = 1 + 16 = 17$

Common Traps & FAQs

Wait, why is the product of the parameters

t_{P} \cdot t_{Q} = - 1

for a focal chord?

This is a beautiful property of parabolas! If you write the equation of a line passing through two points

P (a t_{P}^{2}, 2 a t_{P})

and

Q (a t_{Q}^{2}, 2 a t_{Q})

, the slope is

\frac{2 a ( t _{Q} - t _{P} )}{a ( t _{Q}^{2} - t _{P}^{2} )} = \frac{2}{t _{P} + t _{Q}}

. If this line passes through the focus

(a, 0)

, the slope must also be

\frac{2 a t _{P} - 0}{a t _{P}^{2} - a} = \frac{2 t _{P}}{t _{P}^{2} - 1}

. Equating these two expressions for the slope leads directly to

t_{P} \cdot t_{Q} = - 1

. It is a powerful shortcut that saves you from tedious coordinate geometry!

Can I use the slope formula instead of the section formula to find the ratio?

Absolutely! Since the focus

S

, point

P

, and point

Q

are collinear, the slope of

S P

must equal the slope of

S Q

. You can set

\frac{y _{S} - y _{P}}{x _{S} - x _{P}} = \frac{y _{Q} - y _{S}}{x _{Q} - x _{S}}

. This will give you the same ratio

m : n

. Both methods are valid, but the section formula is often more direct when you are already working with coordinates.

Why did we use the x-coordinate for the section formula? Could we have used the y-coordinate?

You can definitely use the y-coordinate! The section formula applies to both coordinates independently. Using the y-coordinate, you would have

y_{S} = \frac{m \cdot y _{Q} + n \cdot y _{P}}{m + n}

. Since

y_{S} = 0

, this simplifies to

0 = m \cdot y_{Q} + n \cdot y_{P}

, or

\frac{m}{n} = - \frac{y _{P}}{y _{Q}}

. This is actually even faster! Just be careful with the signs.

Animated Solution for Mathematics - Conic Sections: Let the point $P$ of the focal chord $P Q$ of the parabola $y^{2} = 16 x$ be $(1, - 4)$ . If the focus of the parabola divides the chord $P Q$ in the ratio $m : n$ , $gcd (m, n) = 1$ , then $m^{2} + n^{2}$ is equal to :

Select Answer:

Visualized Solution

Equation of the Parabola

Locating the Focus $S$

Parametric Coordinates of $P$

Finding Parameter $t_{P}$

Property of the Focal Chord

Finding Parameter $t_{Q}$

Coordinates of Point $Q$

Calculating Coordinates of $Q$

Applying the Section Formula

Substituting Values into Section Formula

Solving for the Ratio $m : n$

Final Calculation of $m^{2} + n^{2}$

The Sigma Insight: Standard Equations of Parabola, Ellipse, and Hyperbola

Analyzing the Setup

The Parametric Key

The Focal Chord Property

Finding the Other End

The Final Ratio

Similar Questions

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Let the length of the focal chord $P Q$ of the parabola $y^{2} = 12 x$ be 15 units. If the distance of $P Q$ from the origin is $p$ , then $10 p^{2}$ is equal to _______

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If the x-intercept of a focal chord of the parabola $y^{2} = 8 x + 4 y + 4$ is 3, then the length of this chord is equal to

Let $O$ be the vertex of the parabola $x^{2} = 4 y$ and $Q$ be any point on it. Let the locus of the point $P$ , which divides the line segment $O Q$ internally in the ratio $2 : 3$ be the conic $C$ . Then the equation of the chord of $C$ , which is bisected at the point $(1, 2)$ , is :

Select Answer:

Visualized Solution

Equation of the Parabola

Locating the Focus .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } S

Parametric Coordinates of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } P

Finding Parameter .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } tP​

Property of the Focal Chord

Finding Parameter .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } tQ​

Coordinates of Point .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Q

Calculating Coordinates of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Q

Applying the Section Formula

Substituting Values into Section Formula

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

Final Calculation of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } m2+n2

The Sigma Insight: Standard Equations of Parabola, Ellipse, and Hyperbola

Analyzing the Setup

The Parametric Key

The Focal Chord Property

Finding the Other End

The Final Ratio

Similar Questions

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If one end of a focal chord of the parabola, y2=16x is at (1,4), then the length of this focal chord is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let the length of the focal chord PQ of the parabola y2=12x be 15 units. If the distance of PQ from the origin is p, then 10p2 is equal to _______

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let y=mx+c,m>0 be the focal chord of y2=−64x, which is tangent to (x+10)2+y2=4. Then, the value of 42​(m+c) is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the x-intercept of a focal chord of the parabola y2=8x+4y+4 is 3, then the length of this chord is equal to

Locating the Focus $S$

Parametric Coordinates of $P$

Finding Parameter $t_{P}$

Finding Parameter $t_{Q}$

Coordinates of Point $Q$

Calculating Coordinates of $Q$

Solving for the Ratio $m : n$

Final Calculation of $m^{2} + n^{2}$

If one end of a focal chord of the parabola, $y^{2} = 16 x$ is at $(1, 4)$ , then the length of this focal chord is

Let the length of the focal chord $P Q$ of the parabola $y^{2} = 12 x$ be 15 units. If the distance of $P Q$ from the origin is $p$ , then $10 p^{2}$ is equal to _______

Let $y = m x + c, m > 0$ be the focal chord of $y^{2} = - 64 x$ , which is tangent to $(x + 10)^{2} + y^{2} = 4$ . Then, the value of $42 (m + c)$ is equal to

If the x-intercept of a focal chord of the parabola $y^{2} = 8 x + 4 y + 4$ is 3, then the length of this chord is equal to