The axis of a parabola is the line y=x and its vertex and focus are in the first quadrant …

Question

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...

Accepted Answer

Equation of directrix

$$\Rightarrow y=-x$$

By definition of parabola,

$$\begin{aligned}
& P M=P F \
& \left|\frac{1+K}{\sqrt{2}}\right|=\sqrt{(1-2)^2+(K-2)^2} \
& \frac{(1+K)^2}{2}=1+K^2+4-4 K \
& 1+K^2+2 K=10+2 K^2-8 K \
& K^2-10 K+9=0 \
& (K-9)(K-1)=0 \
& \therefore \quad K=1 \text { or } K=9
\end{aligned}$$

Solutions · 1