## Question

Find the shortest and largest distance from the point (2, –7) to the circle

*x*^{2} + *y*^{2} – 14*x* – 10*y* – 151 = 0

### Solution

Shortest distance = 2 & Largest distance = 28

Let *S* ≡ *x*^{2} + *y*^{2} – 14*x* – 10*y* – 151 = 0

∴ *S*_{1} = (2)^{2} + (–7)^{2} – 14(2) – 10(–7) – 151

= 56 < 0

∴ *P*(2, –7) inside the circle

∴ **Shortest distance** = *PA* = *r* – *CP* = 15 – 13 = 2

and **Largest distance** = PB = *r* = *CP* = 15 + 13 = 28

#### SIMILAR QUESTIONS

Find the equation of the circle passing through the three non-collinear points (1, 1), (2, –1) and (3, 2).

Show that the four points (1, 0), (2, –7), (8, 1) and (9, –6) are concyclic.

Find the equation of the circle whose diameter is the line joining the points (–4, 3) and (12, –1). Find also the intercept made by it on *y*-axis.

Find the equation of the circle which touches the axis of *y* at a distance of 4 units from the origin and cuts the intercept of 6 units from the axis of *x*.

Find the equation of the circle which passes through the origin and makes intercepts of length *a* and *b* on the *x* and *y* axes respectively.

Find the equation of the circle which touches the axes and whose centre lies on the line *x* – 2*y* = 3.

A circle of radius 2 lies in the first quadrant and touches both the axes of co-ordinates. Find the equation of the circle with centre at (6, 5) and touching the above circle externally.

A circle of radius 5 units touches the co-ordinates axes in first quadrant. If the circle makes one complete roll on *x*-axis along the positive direction of *x*-axis, find its equation in the new position.

Discuss the position of the points (1, 2) and (6, 0) with respect to the circle

*x*^{2} + *y*^{2} – 4*x* +2*y* – 11 = 0.

Find the points of intersection of the line 2*x* + 3*y* = 18 and the circle *x*^{2} +*y*^{2} = 25.