Tangents to Circles in Analytical Geometry

≈ 25 min

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Tangents to Circles in Analytical Geometry

At a point on a circle, the tangent is perpendicular to the radius. Find the radius gradient from centre to contact point, take the negative reciprocal for the tangent, then use point-gradient form.

Worked reasoning

The radius from $(0,0)$ to $(3,4)$ has gradient $4/3$ .
The tangent gradient is $-3/4$ .
$y-4=-\tfrac34(x-3)$ , so $y=-3x/4+25/4$ .

Key idea

The tangent gradient is the negative reciprocal of the non-axis-aligned radius gradient.

Find the tangent line to $x^2+y^2=25$ at $(3,4)$ .

Which statement best captures the central mathematical idea in Tangents to Circles in Analytical Geometry?

The tangent gradient is the negative reciprocal of the non-axis-aligned radius gradient.

The tangent and radius have equal gradients because they meet at one point.

A memorised formula is valid even when its conditions are not met.

The diagram or notation replaces the need to justify each mathematical step.

When starting a problem about Tangents to Circles in Analytical Geometry, which move is most reliable?

Find the centre and contact point, verify contact, then calculate the radius gradient.

Round every value before identifying the mathematical structure.

Choose the longest available formula and substitute immediately.

Assume the desired conclusion and work backwards without checking equivalence.

Which statement is a misconception that must be rejected when working with Tangents to Circles in Analytical Geometry?

The tangent and radius have equal gradients because they meet at one point.

The tangent gradient is the negative reciprocal of the non-axis-aligned radius gradient.

Find the centre and contact point, verify contact, then calculate the radius gradient.

Check the gradient product is $-1$ and the contact point satisfies both equations.

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