Question

PQ is tangent to a circle with centre O. If \(OQ = a\), \(OP = a + 2\) and \(PQ = 2b\), then relation between \(a\) and \(b\) is

• A. \(a^2 + (a + 2)^2 = (2b)^2\)
• B. \(b^2 = a + 4\)
• C. \(2a^2 + 1 = b^2\)
• D. \(b^2 = a + 1\)

Hint:

Always draw the radius to the point of contact of a tangent to identify the right-angled triangle and apply the Pythagorean property.

Answer 1

The Correct Option is D

To find the relationship between \(a\) and \(b\), consider the geometry of the problem involving the tangent to a circle.

Given:

• The tangent \(PQ\) to the circle at point \(Q\) has length \(2b\).
• The radius \(OQ\) of the circle is \(a\).
• The distance \(OP\) from the center to the point \(P\) is \(a + 2\).

By the Pythagorean theorem, since \(PQ\) is tangent to the circle at \(Q\), and \(OQ\) is the radius, we have the perpendicular relationship:

The right triangle \(OPQ\) has:

• Hypotenuse: \(OP = a + 2\).
• One leg (radius): \(OQ = a\).
• Other leg (tangent): \(PQ = 2b\).

Applying the Pythagorean theorem:

\(((a + 2)^2 = a^2 + (2b)^2)\)

Let's simplify and solve:

1. Expand the equation: \((a + 2)^2 = a^2 + 4b^2\)
2. Simplify the left side: \(a^2 + 4a + 4 = a^2 + 4b^2\)
3. Subtract \(a^2\) from both sides: \(4a + 4 = 4b^2\)
4. Divide by 4 on both sides: \(a + 1 = b^2\)

Thus, the relationship between \(a\) and \(b\) is:

\(b^2 = a + 1\)

Therefore, the correct answer is: \(b^2 = a + 1\).