Let the point $ P $ of the focal chord $ PQ $ of the parabola $ y^2 = 16x $ be $ (1, -4) $. If the focus of the parabola divides the chord $ PQ $ in the ratio $ m : n $, gcd($m, n$) = 1, then $ m^2 + n^2 $ is equal to:
Show Hint
- 17
- 10
- 37
- 26
The Correct Option is A
Approach Solution - 1
The equation of the parabola is: \[ y^2 = 16x; \, a = 4 \] The focus \( S \) is \( (4, 0) \), and the point \( P \) is \( (1, -4) \).
From the equation of the parabola, we know the parametric equations for the points on the parabola: \[ t_1 = -4, \, 2a t_1 = -4 \implies t_1 = \frac{-1}{2} \] \[ t_2 = 2 \implies Q(at_2^2, 2at_2) = (16, 16) \] Let \( S \) divides \( PQ \) internally in the ratio \( \lambda : 1 \): \[ 16\lambda - 4 = 0 \quad \implies \lambda = \frac{1}{4} \] Thus, the ratio \( \frac{m}{n} = \frac{1}{4} \), and: \[ m^2 + n^2 = 1 + 16 = 17 \]
Approach Solution -2
The problem provides a parabola \( y^2 = 16x \) and one endpoint \( P(1, -4) \) of a focal chord \( PQ \). We need to find the value of \( m^2 + n^2 \), where \( m:n \) is the ratio in which the focus divides the chord \( PQ \), and \( \text{gcd}(m, n) = 1 \).
Concept Used:
1. Standard Parabola Equation: The equation of a standard parabola is \( y^2 = 4ax \). Its focus is at \( F(a, 0) \).
2. Parametric Coordinates: Any point on the parabola \( y^2 = 4ax \) can be represented by the parametric coordinates \( (at^2, 2at) \), where \( t \) is the parameter.
3. Focal Chord Property: If \( P(at_1^2, 2at_1) \) and \( Q(at_2^2, 2at_2) \) are the endpoints of a focal chord, then the product of their parameters is \( t_1 t_2 = -1 \).
4. Section Formula: If a point \( F(x, y) \) divides the line segment joining \( P(x_1, y_1) \) and \( Q(x_2, y_2) \) in the ratio \( m:n \), the coordinates of \( F \) are given by \( \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \). Alternatively, the ratio can be found by calculating the lengths \( PF \) and \( FQ \) using the distance formula: \( \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \).
Step-by-Step Solution:
Step 1: Identify the parameters of the parabola and its focus.
The given equation of the parabola is \( y^2 = 16x \). Comparing this with the standard form \( y^2 = 4ax \), we get:
\[ 4a = 16 \implies a = 4 \]
The focus \( F \) of the parabola is at \( (a, 0) \), which is \( F(4, 0) \).
Step 2: Find the parameter \( t_1 \) corresponding to the point \( P(1, -4) \).
Let the coordinates of \( P \) be \( (at_1^2, 2at_1) \). Substituting \( a = 4 \), we have \( P(4t_1^2, 8t_1) \).
\[ (4t_1^2, 8t_1) = (1, -4) \]
Equating the y-coordinates:
\[ 8t_1 = -4 \implies t_1 = -\frac{4}{8} = -\frac{1}{2} \]
We can verify this with the x-coordinate: \( 4t_1^2 = 4 \left(-\frac{1}{2}\right)^2 = 4 \left(\frac{1}{4}\right) = 1 \). This is correct.
Step 3: Find the parameter \( t_2 \) for the other end of the focal chord, \( Q \).
Using the property of a focal chord, \( t_1 t_2 = -1 \):
\[ \left(-\frac{1}{2}\right) t_2 = -1 \implies t_2 = 2 \]
Step 4: Determine the coordinates of the point \( Q \).
The coordinates of \( Q \) are \( (at_2^2, 2at_2) \). Substituting \( a = 4 \) and \( t_2 = 2 \):
\[ x_Q = 4(2)^2 = 16 \] \[ y_Q = 2(4)(2) = 16 \]
So, the coordinates of point \( Q \) are \( (16, 16) \).
Step 5: Calculate the ratio \( m:n \) in which the focus \( F(4, 0) \) divides the chord \( PQ \).
The ratio is \( m:n = PF : FQ \). We use the distance formula to find the lengths of the segments \( PF \) and \( FQ \).
Coordinates are: \( P(1, -4) \), \( F(4, 0) \), and \( Q(16, 16) \).
\[ PF = \sqrt{(4-1)^2 + (0 - (-4))^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \] \[ FQ = \sqrt{(16-4)^2 + (16-0)^2} = \sqrt{12^2 + 16^2} = \sqrt{144 + 256} = \sqrt{400} = 20 \]
The ratio \( m:n \) is:
\[ \frac{m}{n} = \frac{PF}{FQ} = \frac{5}{20} = \frac{1}{4} \]
Since \( \text{gcd}(1, 4) = 1 \), we have \( m = 1 \) and \( n = 4 \).
Final Computation & Result:
We need to find the value of \( m^2 + n^2 \).
\[ m^2 + n^2 = (1)^2 + (4)^2 = 1 + 16 = 17 \]
Thus, the value of \( m^2 + n^2 \) is 17.
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