The axis of a parabola is the line y = x and its vertex and focus are in the first quadrant at distances and units from the origin, respectively. If the point (1, k) lies on the parabola, then a possible value of k is : [2025]

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Solution

Q.
The axis of a parabola is the line y = x and its vertex and focus are in the first quadrant at distances $\sqrt{2}$ and $2 \sqrt{2}$ units from the origin, respectively. If the point (1, k) lies on the parabola, then a possible value of k is : [2025]

1 8

2 4

3 3

4 9

Ans.
(4)

For the parabola, axis is y = x

$\Rightarrow$ Directrix is x + y = 0

Also, vertex and focus are of $\sqrt{2}$ and $2 \sqrt{2}$ from origin

$\Rightarrow$ Vertex is (1, 1) and focus = (2, 2)

The point P(1, k) lies on parabola.

Using definition of parabola

PS = PM

$\Rightarrow \sqrt{{(1 - 2)}^{2} + {(k - 2)}^{2}} = \frac{1 + k}{\sqrt{2}}$

$\Rightarrow k^{2} - 10 k + 9 = 0$

$\Rightarrow (k - 9) (k - 1) = 0$

$∴$ k = 9 and k = 1.

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