wiz-icon
MyQuestionIcon
MyQuestionIcon
1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

There is a circle on the co -ordinate plane with two tangent lines x - y = 3 and x - y = 0. The meeting points of the circle with the 2 lines are (32,32) and (0,- 3). Find the radius and the centre of the circle.


A

(3/4)√10,(3/4, -3/4)

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
B

9√2, (3/2, 3/4)

No worries! We‘ve got your back. Try BYJU‘S free classes today!
C

3√2, (3/4, 9/4)

No worries! We‘ve got your back. Try BYJU‘S free classes today!
D

4.5/√2 , (3/4, 9/2)

No worries! We‘ve got your back. Try BYJU‘S free classes today!
Open in App
Solution

The correct option is A

(3/4)√10,(3/4, -3/4)


Let us first look at the nature of the 2 lines. We observe that the coefficients of x and y for both the lines in equations are same. But the constant term is different for both terms.

Let us use the concepts in linear equations in 2 variables.

Let us consider 2 lines with equations given below.

The above condition is satisfied by the 2 lines given in the question. This means that the 2 lines are parallel.

In other words, we have 2 parallel tangents to a circle.

This can be represented by the diagram below.

Let PQ and XY be the 2 tangents. A and B are the 2 points of contact with the circle respectively. The centre of the circle is O. Join OA and OB.

Tangents are perpendicular to radii at a point of contact. Tangents PQ and XY are perpendicular to radii OA and OB respectively. So PAO and XBO are of 90.

2 lines that are perpendicular to 2 parallel lines are at 180 to each other. Line segments OA and OB are perpendicular to parallel lines PQ and XY. So, OA and OB are at 180. Thus, AOB is at180 and AOB is a straight line segment.

AOB is a line segment passing centre O. Also A and B are points on the circumference. Thus, AB is the diameter of a circle and its midpoint is centre O.

We already know the coordinates of contact points A and B. The midpoint of A and B will give us the centre. The distance AB is the diameter. So half of distance AB is the radius.

A = (32,32)

B= (0,-3)

Centre is midpoint of AB = (34,34)

Applying distance formula

AB= [(x2x1)2 + (y2y1)2]

AB = [(320)2 + (32+3)2 ]

AB = 3210

Radius is half of AB = 3410

The midpoint of 2 points in coordinate geometry is


flag
Suggest Corrections
thumbs-up
0
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
Introduction
MATHEMATICS
Watch in App
Join BYJU'S Learning Program
CrossIcon