Sunday Afternoon Maths XVIII

Here's this week's collection. Answers & extensions tomorrow. Why not discuss the problems on Twitter using #SundayAfternoonMaths or on Reddit.

Parabola

Source: Alex Through the Looking-Glass: How Life Reflects Numbers and Numbers Reflect Life by Alex Bellos

On a graph of \(y=x^2\), two lines are drawn at \(x=a\) and \(x=-b\) (for \(a,b>0\). The points where these lines intersect the parabola are connected.

What is the y-coordinate of the point where this line intersects the y-axis?

Seven Digits

Source: My Best Puzzles in Mathematics by Hubert Phillips

"I'm thinking of a number. I've squared it. I've squared the square. And I've multiplied the second square by the original number. So I now have a number of seven digits whose final digit is a 7," said Dr. Dingo to his daughter.

Can you work out Dr. Dingo's number?