Question

If $x^{2}-ax-21=0$ and $x^{2}-3ax+35=0$ with $a>0$ have a common root, then $a$ equals:

• A. 1
• B. 2
• C. 4
• D. 5

Hint:

For two quadratics with a common root, subtract to eliminate $x^2$ and solve for the root in terms of parameters; then substitute back.

Answer 1

The Correct Option is C

Step 1: For \(x^2-ax-21=0\) and \(x^2-3ax+35=0\) with common root \(r\): \[ r=\frac{b_1c_2-b_2c_1}{c_1a_2-c_2a_1}=\frac{(-a)(35)-(-3a)(-21)}{(-21)-(35)}=\frac{7a}{4}. \]

Step 2: Also \[ r=\frac{c_1a_2-c_2a_1}{a_1b_2-a_2b_1}=\frac{-56}{-2a}=\frac{28}{a}. \]

Step 3: Equate the two values of \(r\): \[ \frac{7a}{4}=\frac{28}{a}\;\Rightarrow\;a^2=16\;\Rightarrow\;\boxed{a=4} \]