Exercise 387Calculus › Single-variable calculus › Differentiation › Differentiable functionsHigh schoolAdded 2 Nov 2026

Tangent lines through an external point

#tangent line#parabola#geometry

Statement

Find all tangent lines to the parabola $y=x^2$ that pass through the point $A=(1,-3)$ .

Hint

Write the tangent at an unknown abscissa $a$ , then require that it passes through $A$ . The unknown is $a$ , not the slope.

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Tangent at $a$ . For $f(x)=x^2$ , $f'(a)=2a$ , so the tangent at the point of abscissa $a$ is
$y=a^2+2a(x-a)=2ax-a^2.$

Passing through $A$ . We need $-3=2a-a^2$ , that is $a^2-2a-3=0$ , i.e. $(a-3)(a+1)=0$ . Hence $a=3$ or $a=-1$ .

The two tangents.
- $a=3$ : $y=6x-9$ ,
- $a=-1$ : $y=-2x-1$ .