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 :

The correct answer is Option 2. Full mathematical solution steps and hints are accessible in the interactive StemCET workspace.

Maths Q11 – JEE-MAIN 2025

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Home Maths Conic SectionsQuestion #132

Let the point P of the focal chord PQ of the parabola

y^{2} = 16 x

(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 :

Clue 1 of 3: Parabola Properties

Identify the standard form of the parabola $y^{2} = 4 a x$ and determine the coordinates of its focus $(a, 0)$ .

🧠 Full Solution Path

Step 1: Identify Parabola Parameters and Focus✦ Active

The given parabola equation is $y^{2} = 16 x$ . Comparing this with the standard form $y^{2} = 4 a x$ , we find $4 a = 16$ , which implies $a = 4$ . Therefore, the focus of the parabola, $S$ , is at $(a, 0) = (4, 0)$ .

Step 2: Find Parametric Coordinates of P and Q○ Expand

Let the parametric coordinates of point P be $(a t_{1}^{2}, 2 a t_{1})$ . Given $P = (1, - 4)$ and $a = 4$ , we have $4 t_{1}^{2} = 1 ⟹ t_{1}^{2} = \frac{1}{4} ⟹ t_{1} = \pm \frac{1}{2}$ . Also, $2 (4) t_{1} = - 4 ⟹ 8 t_{1} = - 4 ⟹ t_{1} = - \frac{1}{2}$ . Thus, $t_{1} = - \frac{1}{2}$ . For a focal chord, the product of the parameters is $t_{1} t_{2} = - 1$ . Substituting $t_{1} = - \frac{1}{2}$ , we get $(- \frac{1}{2}) t_{2} = - 1 ⟹ t_{2} = 2$ . The coordinates of point Q are $(a t_{2}^{2}, 2 a t_{2}) = (4 (2^{2}), 2 (4) (2)) = (16, 16)$ .

💡 Teacher's Secret Hint

Remember the condition $t_{1} t_{2} = - 1$ for a focal chord passing through the focus.

Step 3: Apply Section Formula and Calculate○ Expand

The focus $S (4, 0)$ divides the chord PQ, with $P (1, - 4)$ and $Q (16, 16)$ , in the ratio $m : n$ . Using the section formula for the x-coordinate:

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

This simplifies to $4 (m + n) = 16 m + n ⟹ 4 m + 4 n = 16 m + n ⟹ 3 n = 12 m ⟹ n = 4 m$ . The ratio $m : n = 1 : 4$ . Since $gcd (m, n) = 1$ , we have $m = 1$ and $n = 4$ . Finally, we need to calculate $m^{2} + n^{2}$ :

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

💡 Teacher's Secret Hint

Ensure the ratio $m : n$ is in its simplest form as $gcd (m, n) = 1$ is given.

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Maths Question 11 – JEE-MAIN 2025

Clue 1 of 3: Parabola Properties

🧠 Full Solution Path