(a) Show that one possible value of $a$ is and the other possible value is

Key takeaway: This is another tangent-to-circle setup, but the pasted working contains several transcription errors. The reliable method is still the same: substitute the line into the circle, obtain a quadratic, and impose discriminant $0$ for tangency.

The working shown in the prompt has several inconsistent lines, so the clean method is to use the tangent condition directly.

Step 1: Substitute the line into the circle

If a line is tangent to a circle, substituting the line equation into the circle equation produces a quadratic in one variable.

Step 2: Use the tangency condition

Because the line touches the circle at exactly one point, that quadratic must have one repeated root. Therefore,

$b^2-4ac=0.$

Step 3: Solve the resulting equation

After simplifying the coefficients correctly, solve the resulting quadratic in $a$ to get the two possible values.

Important note

Several expressions in the pasted working do not match each other, so the exact numeric answers cannot be trusted from the transcription alone. The intended method is the discriminant test above.

Pitfalls the pros know 👇 Do not mix coefficients from different lines of working. In the pasted solution, the terms involving $x^2$, $x$, and the constant term change from line to line, which changes the final equation for $a$. When copying algebra from a scan or photo, re-check every expansion before solving.

What if the problem changes? If the line equation or circle equation changes slightly, the method is unchanged: substitute, expand, set the discriminant to $0$, and solve for the parameter. If the line is not tangent but secant, the discriminant is positive; if it misses the circle, it is negative.

Tags: tangent condition, discriminant, quadratic equation