Let the image of parabola x 2 = 4 y , in the line x - y = 1 be ( y + a ) 2 = b ( x - c ) , a , b , c . Then a + b + c is equal to [2026]

Question

Let the image of parabola $x^{2} = 4 y$ , in the line $x - y = 1$ be ${(y + a)}^{2} = b (x - c),$ $a, b, c \in ℕ$ . Then $a + b + c$ is equal to [2026]

Smart Achievers · Accepted Answer

(4)

$Parametric point P on x^{2} = 4 y is P (2 t, t^{2})$

$∴$ $mirror image of P in x - y = 1 is$

$Q \equiv (2 t - \frac{2.1 . (2 t - t^{2} - 1)}{2}, t^{2} + \frac{2.2 (1) . (2 t - t^{2} - 1)}{2})$

$Q \equiv (t^{2} + 1, 2 t - 1) \equiv (h, k)$

$∴$ $locus of Q is x = \frac{{(y + 1)}^{2}}{4} + 1$ which is the required parabola.

$∴$ ${(y + 1)}^{2} = 4 (x - 1)$

$∴$ $a = 1, b = 4, c = 1$

$∴$ $a + b + c = 6$

Solution

Q.
Let the image of parabola $x^{2} = 4 y$ , in the line $x - y = 1$ be ${(y + a)}^{2} = b (x - c),$ $a, b, c \in ℕ$ . Then $a + b + c$ is equal to [2026]

1 4

2 12

3 8

4 6

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Solution

Q. Let the image of parabola x2=4y, in the line x-y=1 be (y+a)2=b(x-c), a,b,c∈ℕ. Then a+b+c is equal to [2026]

1 4

2 12

3 8

4 6

Ans. (4) Parametric point P on x2=4y is P(2t,t2) ∴ mirror image of P in x-y=1 is Q≡(2t-2.1.(2t-t2-1)2, t2+2.2(1).(2t-t2-1)2) Q≡(t2+1, 2t-1)≡(h,k) ∴ locus of Q is x=(y+1)24+1 which is the required parabola. ∴ (y+1)2=4(x-1) ∴ a=1, b=4, c=1 ∴ a+b+c=6

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Q.
Let the image of parabola $x^{2} = 4 y$ , in the line $x - y = 1$ be ${(y + a)}^{2} = b (x - c),$ $a, b, c \in ℕ$ . Then $a + b + c$ is equal to [2026]